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Full text of "Vector calculus, with applications to physics"

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VECTOR CALCULUS 



WITH APPLICATIONS TO PHYSICS 



BY 



JAMES BYRNIE SHAW 



PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS 



ILLUSTRATED 




NEW YORK 
D. VAN NOSTRAND COMPANY 

Eight Warren Street 
1922 



s 



Copyright, 1922 
By D. Van Nostrand Company 



All rights reserved, including that of translation into 
foreign languages, including the Scandinavian 



Printed in the United States of America 



PREFACE. 

This volume embodies the lectures given on the subject 
to graduate students over a period of four repetitions. The 
point of view is the result of many years of consideration 
of the whole field. The author has examined the various 
methods that go under the name of Vector, and finds that 
for all purposes of the physicist and for most of those of the 
geometer, the use of quaternions is by far the simplest in 
theory and in practice. The various points of view are 
mentioned in the introduction, and it is hoped that the es- 
sential differences are brought out. The tables of com- 
parative notation scattered through the text will assist in 
following the other methods. 

The place of vector work according to the author is in 
the general field of associative algebra, and every method so 
far proposed can be easily shown to be an imperfect form 
of associative algebra. From this standpoint the various 
discussions as to the fundamental principles may be under- 
stood. As far as the mere notations go, there is not much 
difference save in the actual characters employed. These 
have assumed a somewhat national character. It is un- 
fortunate that so many exist. 

The attempt in this book has been to give a text to the 
mathematical student on the one hand, in which every 
physical term beyond mere elementary teims is carefully 
defined. On the other hand for the physical student there 
will be found a large collection of examples and exercises 
which will show him the utility of the mathematical meth- 
ods. So very little exists in the numerous treatments of 
the day that does this, and so much that is labeled vector 

iii 

505384 



IV PREFACE 

analysis is merely a kind of short-hand, that it has seemed 
very desirable to show clearly the actual use of vectors as 
vectors. It will be rarely the case in the text that any use 
of the components of vectors will be found. The triplexes 
in other texts are very seldom much different fiom the ordi- 
nary Cartesian forms, and not worth learning as methods. 

The difficulty the author has found with other texts is 
that after a few very elementary notions, the mathematical 
student (and we may add the physical student) is suddenly 
plunged into the profundities of mathematical physics, as 
if he were familiar with them. This is rarely the case, and 
the object of this text is to make him familiar with them 
by easy gradations. 

It is not to be expected that the book will be free from 
errors, and the author will esteem it a favor to have all 
errors and oversights brought to his attention. He desires 
to thank specially Dr. C. F. Green, of the University of 
Illinois, for his careful assistance in reading the proof, and 
for other useful suggestions. Finally he has gathered his 
material widely, and is in debt to many authors for it, to all 
of whom he presents his thanks. 

James Byrnie Shaw. 
Urbana, III., 
July, 1922. 



TABLE OF CONTENTS. 

Chapter I. Introduction 1 

Chapter II. Scalar Fields 18 

Chapter III. Vector Fields 23 

Chapter IV. Addition of Vectors 52 

Chapter V. Vectors in a Plane 62 

Chapter VI. Vectors in Space 94 

Chapter VII. Applications 127 

1. The Scalar of two Vectois 127 

2. The Vector of two Vectors 136 

3. The Scalar of three Vectors 142 

4. The Vector of three Vectors 143 

Chapter VIII. Differentials and Integrals 145 

1. Differentiation as to one Scalar Parameter .... 145 

Two Parameters 151 

2. Differentiation as to a Vector 155 

3. Integration 196 

Chapter IX. The Linear Vector Function 218 

Chapter X. Deformable Bodies 253 

Strain 253 

Kinematics of Displacement 265 

Stress 269 

Chapter XL Hydrodynamics 287 



VECTOR CALCULUS 



CHAPTER I 
INTRODUCTION 
1. Vector Calculus. By this term is meant a system of 
mathematical thinking which makes use of a special class 
of symbols and their combinations according to certain 
given laws, to study the mathematical conclusions resulting 
from data which depend upon geometric entities called 
vectors, or physical entities representable by vectors, or 
more generally entities of any kind which could be repre- 
sented for the purposes under discussion by vectors. These 
vectors may be in space of two or three or even four or 
more dimensions. A geometric vector is a directed segment 
of a straight line. It has length (including zero) and direc- 
tion. This is equivalent to saying that it cannot be de- 
fined merely by one single numerical value. Any problem 
of mathematics dependent upon several variables becomes 
properly a problem in vector calculus. For instance, 
analytical geometry is a crude kind of vector calculus. 

Several systems of vector calculus have been devised, differing in 
their fundamental notions, their notation, and their laws of combining 
the symbols. The lack of a uniform. notation is deplorable, but there 
seems little hope of the adoption of any uniform system soon. Existing 
systems have been rather ardently promoted by mathematicians of the 
same nationality as their authors, and disagreement exists as to their 
relative simplicity, their relative directness, and their relative logical 
exactness. These disagreements arise sometimes merely with regard 
to the proper manner of representing certain combinations of the 
symbols, or other matters which are purely matters of convention; 

1 



2 YKCTOR CALCULUS 

sometimes they are due to different views as to what are the import an1 
things to find expressions for; and sometimes they are due to more 
fundamental divergences of opinion as to the real character of the 
mathematical ideas underlying any system of this sort. We will in- 
dicate these differences and dispose of them in this work. 

2. Bases. We may classify broadly the various systems 
of vector calculus as geometric and algebraic. The former 
is to be found wherever the desire is to lay emphasis on the 
spatial character of the entities we are discussing, such as 
the line, the point, portions of a plane, etc. The latter 
lays emphasis on the purely algebraic character of the 
entities with which the calculations are made, these entities 
being similar to the positive and negative, and the imag- 
inary of ordinary algebra. For the geometric vector 
systems, the symbolism of the calculus is really nothing 
more than a short-hand to enable one to follow certain 
operations upon real geometric elements, with the possi- 
bility kept always in mind that these entities and the 
operations may at any moment be called to the front to 
take the place of their short-hand representatives. For 
the algebraic systems, the symbolism has to do with 
hypernumbers, that is, extensions of the algebraic negative 
and imaginary numbers, and does not pretend to be the 
translation of actual operations which can be made visible, 
any more than an ordinary calculation of algebra could be 
paralleled by actual geometric or physical operations. 

If these distinctions are kept in mind the different points 
of view become intelligible. The best examples of geo- 
metric systems are the Science of Extension of Grassmann, 
with its various later forms, the Geometry of Bynames of 
Study, the Geometry of Lines of Saussure, and the Geometry 
of Feuillets of Cailler. The best examples of algebraic 
systems are the Quaternions of Hamilton, Dyadics of Gibbs, 



INTRODUCTION ,3 

Multenions of McAulay, Biquaternions of Clifford, Tri- 

quaternions of Combebiac, Linear Associative Algebra of 

Peirce. Various modifications of these exist, and some 

mixed systems may be found, which will be noted in the 

proper places. 

The idea of using a calculus of symbols for writing out geometric 
theorems perhaps originated with Leibniz, 1 though what he had in 
mind had nothing to do with vector calculus in its modern sense. The 
first effective algebraic vector calculus was the Quaternions of Hamilton 2 
(1843), the first effective geometric vector calculus was the Ausdehn- 
ungslehre of Grassmann 3 (1844). They had predecessors worthy of 
mention and some of these will be noticed. 

3. Hypernumbers. The real beginning of Vector Cal- 
culus was the early attempt to extend the idea of number. 
The original theory of irrational number was metric, 4 and 
defined irrationals by means of the segments of straight 
lines. When to this was added the idea of direction, so 
that the segments became directed segments, what we now 
call vectors, the numbers defined were not only capable of 
being irrational, but they also possessed quality, and could 
be negative or positive. Ordinary algebra is thus the first 
vector calculus. If we consider segments with direction 
in a plane or in space of three dimensions, then we may call 
the numbers they define hypernumbers. The source of the 
idea was the attempt to interpret the imaginary which 
had been created to furnish solutions for any quadratic or 
cubic. The imaginary appears early in Cardan's work. 5 
For instance he gives as solution of the problem of separating 
10 into two parts whose product is 40, the values 
5 + V — 15, and 5 — V — 15. He considered these 
numbers as impossible and of no use. Later it was dis- 
covered that in the solution of the cubic by Cardan's 
formula there appeared the sum of two of these impossible 



4 VWCTOfl CALCULUS 

values when the answer actually was real. Bombelli #;ive 
as the solution of the cubic r 3 = 15x + 4 the form 

^(2 + V - 121) + ^(2 - V - 121) = 4. 

These impossible numbers incited much thought and 
there came about several attempts to account for them and 
to interpret them. The underlying question was essen- 
tially that of existence, which at that time was usually 
sought for in concrete cases. The real objection to the 
negative number was its inapplicability to objects. Its 
use in a debit and credit account would in this sense give it 
existence. Likewise the imaginary and the complex num- 
ber, and later others, needed interpretation, that is, applica- 
tion to physical entities. 

4. Wessel, a Danish surveyor, in 1797, produced a 
satisfactory method 7 of defining complex numbers by means 
of vectors in a plane. This same method was later given 
by Argand 8 and afterwards by Gauss 9 in connection with 
various applications. Wessel undertook to go farther and 
in an analogous manner define hypernumbers by means 
of directed segments, or vectors, in space of three dimen- 
sions. He narrowly missed the invention of quaternions. 
In 1813 Servois 10 raised the question whether such vectors 
might not define hypernumbers of the form 

. p cos a + q cos (3 + r cos y 

and inquired what kind of non-reals p, q, r would be. He 
did not answer the question, however, and Wessel's paper 
remained unnoticed for a century. 

5. Hamilton gave the answer to the question of Servois 
as the result of a long investigation of the whole problem. 11 
He first considered algebraic couples, that is to say in our 
terminology, hypernumbers needing two ordinary numerical 



INTRODUCTION 5 

values to define them, and all possible modes of combining 
them under certain conditions, so as to arrive at a similar 
couple or hypernumber for the product. He then con- 
sidered triples and sets of numbers in general. Since — 1 
and i = V — 1 are roots of unity, he paid most attention 
to definitions that would lead to new roots of unity. 

His fundamental idea is that the couple of numbers (a, b) 
where a and b are any positive or negative numbers, rational 
or irrational, is an entity in itself and is therefore subject 
to laws of combination just as are single numbers. For 
instance, we may combine it with the other couple (x, y) 
in two different ways : 

(a, b) + (x, y) = (a + x, b + y) 

(a, 6) X (x, y) = {ax — by, ay + bx). 

In the first case we say we have, added the couples, in the 
second case that we have multiplied them. It is possible 
to define division also. In both cases if we set the couple 
on the right hand side equal to {u, v) we find that 

dujdx — dv/dy, dujdy = — dv/dx. 

Pairs of functions u, v which satisfy these partial differential 
equations Hamilton called conjugate functions. The partial 
differential equations were first given by Cauchy in this 
connection. The particular couples 

€l = (1, 0), € 2 = (0, 1) 

play a special role in the development, for, in the first 
place, any couple may be written in the form 

(a, b) = aei + be 2 

and the notation of couples becomes superfluous; in the 
second place, by defining the products of ei and e 2 in various 
ways we arrive at various algebras of couples. The general 



C> VECTOR CALCULUS 

definition would be, using the • for X, 

€l'€i = Cin€i + Cii 2 € 2 , €i'€ 2 = Ci2i€i + ^12262, 

€2'€i = C2ll€i + C212€2> «2 * €2 = C221«l + C222€2- 

By varying the choice of the arbitrary constants c, and 
Hamilton considered several different cases, different 
algebras of couples could be produced. In the case above 
the c's are all zero except 

Cm = 1, C122 = 1, C212 — 1, C221 = — 1. 

From the character of 4 it may be regarded as entirely 
identical with ordinary 1, and it follows therefore that 
e 2 may be regarded as identical with the V — 1. On the 
other hand we may consider €1 to be a unit vector pointing 
to the right in the plane of vectors, and c 2 to be a unit 
vector perpendicular to ei. We have then a vector calculus 
practically identical with Wessel's. The great merit of 
Hamilton's investigation lies of course in its generality. 
He continued the study of couples by a similar study of 
triples and then quadruples, arriving thus at Quaternions. 
His chief difference in point of view from those who followed 
him and who used the concept of couple, triple, etc. {Mul- 
tiple we will say for the general case), is that he invariably 
defined one product, whereas others define usually several. 
6. Multiples. There is a considerable tendency in the 
current literature of vector calculus to use the notion of 
multiple. A vector is usually designated by a triple as 
(x, y, z), and usually such triple is called a vector. It is 
generally tacitly understood that the dimensions of the 
numbers of the triple are the same, and in fact most of the 
products defined would have no meaning unless this 
homogeneity of dimension were assumed to hold. We 
find products defined arbitrarily in several ways. For 
instance, the scalar product of the triples (a, b, c) and (x, y, z) 



INTRODUCTION 7 

is =fc (ax + by + cz), the sign depending upon the person 
giving the definition; the vector product of the same two 
triples is usually given as the triple (bz — cy, ex — az, 
ay — bx). It is obvious at once that a great defect of such 
definitions is that the triples involved have no sense until 
the significance of the first number, the second number, 
and the third number in each triple is understood. If 
these depend upon axes for their meaning, then the whole 
calculus is tied down to such axes, unless, as is usually 
done, the expressions used in the definitions are so chosen 
as to be in some respects independent of the particular 
set of axes chosen. When these expressions are thus 
chosen as invariants under given transformations of the 
axes we arrive at certain of the well-known systems of 
vector analysis. The transformations usually selected to 
furnish the profitable expressions are the group of orthog- 
onal transformations. For instance, it was shown by 
Burkhardt 12 that all the invariant expressions or invariant 
triples are combinations of the three following : 

ax + by + cz, 

(bz — cy, ex — az, ay — bx), 

(al + bm + cn)x + (am — bl)y + (an — cl)z, 

(bl — am)x + (al -f- bm + cn)y + (bn — cm)z, 

(cl — ari)x + (cm — bn)u + (al + bm + cn)z. 

A study of vector systems from this point of view has 
been made by Schouten. 13 

7. Quaternions. In his first investigations, Hamilton 
was chiefly concerned with the creation of systems of 
hypernumbers such that each of the defining units, similar 
to the ei and € 2 above, was a root of unity. 14 That is, the 
process of multiplication by iteration would bring back the 
multiplicand. He was actually interested in certain special 



8 VECTOR CALCULUS 

cases of abstract groups, 15 and if he had noticed the group 
property his researches would perhaps have extended into 
the whole field of abstract groups. In quaternions he found 
a set of square roots of — 1, which he designated by i, j, k, 
connected with his triples though belonging to a set of quad- 
ruples. In his Lectures on Quaternions, the first treatise he 
published on the subject, he chose a geometrical method of 
exposition, consequently many have been led to think of 
quaternions as having a geometric origin. However, the 
original memoirs show that they were reached in a purely 
algebraic way, and indeed according to Hamilton's philoso- 
phy were based on steps of time as opposed to geometric 
steps or vectors. 

The geometric definition is quite simple, however, and 
not so abstract as the purely algebraic definition. Ac- 
cording to this idea, numbers have a metric definition, a 
number, or hypernumber, being the ratio of two vectors. 
If the vectors have the same direction we arrive at the 
ordinary numerical scale. If they are opposite we arrive 
at the negative numbers. If neither in the same direction 
nor opposite we have a more general kind of number, a 
hypernumber in fact, which is a quaternion, and of which 
the ordinary numbers and the negative numbers are 
merely special cases. If we agree to consider all vectors 
which are parallel and in the same direction as equivalent, 
that is, call them free vectors, then for every pair of vectors 
from the origin or any fixed point, there is a quaternion. 
Among these quaternions relations will exist, which will 
be one of the objects of study of later chapters. 

8. Mobius was one of the early inventors of a vector 
calculus on the geometric basis. In his Barycentrisch.es 
Kalkul 16 he introduced a method of deriving points from 
other points by a process called addition, and several 



INTRODUCTION 9 

applications were made to geometry. The barycentric 
calculus is somewhat between a system of homogeneous 
coordinates and a real vector calculus. His addition was 
used by Grassmann. 

9. Grassmann in 1844 published his treatise called Die 
lineale Ausdehnungslehre 17 in which several different proc- 
esses called multiplication are used for the derivation of 
geometric entities from other geometric entities. These 
processes make use of a notation which is practically a 
sort of short-hand for the geometric processes involved. 
Grassmann considered these various kinds of multiplication 
abstractly, leaving out of account the meaning of the 
elements multiplied. His methods apply to space of N 
dimensions. In the symmetric multiplication it is possible 
to interchange any two of the factors without affecting the 
result. In the circular multiplication the order may be 
changed cyclically. In the lineal multiplication all the 
laws hold as well for any factors which are linear combina- 
tions of the hypernumbers which define the base, as for 
those called the base. He studies two species of circular 
multiplication. If the defining units of the base are ex, e 2 , e 3 
• • •€„, then we have in the first variety of circular multipli- 
cation the laws 

€l 2 + € 2 2 + 6 3 2 + • • • + € n 2 = 0, €i€j = €j€i. 

In the second variety we have the laws 

ei 2 = 0, e/ = 0, • - • e n 2 = 0, Mi = 0, *+j. 

In the lineal genus of multiplication he studies two 
species, in the first, called the algebraic multiplication, we 
have the law 

My = *fii for all i, j. 

while in the second, called the exterior multiplication, the 
interchange of any two factors changes the sign of the 



10 VECTOR CALCULUS 

result. Of the latter there are two varieties, the progressive 
multiplication in which the number of dimensions of the 
geometric figure which is the product is the sum of the 
dimensions of the factors, while in the other, called re- 
gressive multiplication, the dimension of the product is the 
difference between the sum of the dimensions of the factors 
and N the dimension of the space in which the operation 
takes place. From the two varieties he deduces another 
kind called interior multiplication. 

If we confine our thoughts to space of three dimensions, 
defined by points, and if €1, e 2 , e 3 , e 4 are such points, the 
progressive exterior product of two, as €1, e 2 , is ei€ 2 and 
represents the segment joining them if they do not coincide. 
The product is zero if they coincide. The product of this 
into a third point € 3 is ei€ 2 e 3 and represents the parallelogram 
with edges €162, ei€ 3 and the other two parallel to these 
respectively. If all three points are in a straight line the 
product is zero. The exterior progressive product c 1 e 2 e 3 € 4 
represents the parallelepiped with edges €ie 2 , €ie 3 , €i€ 4 and 
the opposite parallel edges. The regressive exterior product 
of €i€ 2 and €ie 3 € 4 is their common point €1. The regressive 
product of €ie 2 e 3 and €ie 2 € 4 is their common line €ie 2 . The 
complement of €1 is defined to be € 2 e 3 e 4 , and of €i€ 2 is e 3 fct, 
and of €i€ 2 e 3 is € 4 . The interior product of any expression 
and another is the progressive or regressive product of the 
first into the complement of the other. For instance, the 
interior product of €1 and e 2 is the progressive product of 
€1 and €i€ 3 e 4 which vanishes. The interior product of e 2 
and e 2 is the product of e 2 and eie 3 e 4 which is € 2 eie 3 e 4 . The 
interior product of €j€ 2 e 3 and ei€ 4 is the product of €ie 2 e 3 
and € 2 e 3 which would be regressive and be the line e 2 e 3 . 

We have the same kinds of multiplication if the expres- 
sions e are vectors and not points, and they may even be 



INTRODUCTION 1 1 

planes. The interpretation is different, however. It is 
easy to see that Grassmann's ideas do not lend themselves 
readily to numerical application, as they are more closely 
related to the projective transformations of space. In 
fact, when translated, most of the expressions would be 
phrased in terms of intersections, points, lines and planes, 
rather than in terms of distances, angles, areas, etc. 

10. Dyadics were invented by Gibbs, 18 and are of both the 
algebraic and the geometric character. Gibbs has, like 
Hamilton, but one kind of multiplication. If we have 
given two vectors a, (3 from the same point, their dyad is a(3. 
This is to be looked upon as a new entity of two dimensions 
belonging to the point from which the vectors are drawn. 
It is not a plane though it has two dimensions, but is really 
a particular and special kind of dyadic, an entity of two- 
dimensional character, such that in every case it can be 
considered to be the sum of not more than three dyads. 
Gibbs never laid any stress on the geometric existence of 
the dyadic, though he stated definitely that it was to be 
considered as a quantity. His greatest stress, however, 
was upon the operative character of the dyadic, its various 
combinations with vectors being easily interpretable. The 
simplest interpretation is from its use in physics to represent 
strain. 

Gibbs also pushed his vector calculus into space of many 
dimensions, and into triadic and higher forms, most of 
which can be used in the theory of the elasticity of crystals. 
The scalar and vector multiplication he considered as 
functions of the dyadic, rather than as multiplications, 
and there are corresponding functions of triadics and 
higher forms. In this respect his point of view is close to 
that of Hamilton, the difference being in the use of the 
dyadic or the quaternion. 

11. Other forms of vector calculus can be reduced to 
3 



12 VECTOR CALCULUS 

these or to combinations of parts of these. The differences 
are usually in the notations, or in the basis of exposition. 

Notations for One Vector 

Greek letters, Hamilton, Tait, Joly, Gibbs. 

Italics, Grassmann,_Peano, Fehr, Ferraris, Macfarlane. 
Heun writes a, b, c. 

Old English or German letters, Maxwell, Jaumann, Jung, 
Foppl, Lorentz, Gans, Abraham, Bucherer, Fischer, 
Sommerfeld. 

Clarendon type, Heaviside, Gibbs, Wilson, Jahnke, Timer- 
ding, Burali-Forti, Marcolongo. 

Length of a vector 
T ( ), Hamilton, Tait, Joly. 
| | , Gans, Bucherer, Timerding. 
Italic corresponding to the ve ctor letter, Wilson, Jaumann, 

&ing, Fischer, Jahnke. Corresponding small italic, 

Macfarlane. 
Mod. ( ), Peano, Burali-Forti, Marcolongo, Fehr. 

Unit of a vector 
U ( ), Hamilton, Tait, Joly, Peano. 
Clarendon small, Wilson. 
( )i, Bucherer, Fischer. 
Corresponding Greek letter, Macfarlane. 
Some write the vector over the length. 

Square of a vector 
( ) 2 . The square is usually positive except in Quaternions, 
where it is negative. 

Reciprocal 

( ) -1 , Hamilton, Tait, Joly, Jaumann. 

tt , Hamilton, Tait, Joly, Fischer, Bucherer. 



CHAPTER II 

SCALAR FIELDS 
1. Fields. If we consider a given set of elements in 
space, we may have for each element one or more quantities 
determined, which can be properly called functions of the 
element. For instance, at each point in space we may have 
a temperature, or a pressure, or a density, as of the air. 
Or for every loop that we may draw in a given space we 
may have a length, or at some fixed point a potential due 
to the loop. Again, we may have at each point in space 
a velocity which has both direction and length, or an 
electric intensity, or a magnetic intensity. Not to multiply 
examples unnecessarily, we can see that for a given range 
of points, or lines, or other geometric elements, we may 
have a set of quantities, corresponding to the various 
elements of the range, and therefore constituting a function 
of the range, and these quantities may consist of numerical 
values, or of vectors, or of other hypernumbers. When 
they are of a simple numerical character they are called 
scalars, and the function resulting is a scalar function. 
Examples are the density of a fluid at each point, the density 
of a distribution of energy, and similar quantities consisting 
of an amount of some entity per cubic centimeter, or per 
square centimeter, or per centimeter. 

EXAMPLES 
(1) Electricity. The unit of electricity is the coulomb, 
connected with the absolute units by the equations 
1 coulomb = 3 • 10° electrostatic units 

== 10 -1 electromagnetic units. 
13 



14 VECTOR CALCULUS 

The density of electricity is its amount in a given volume, 
area, op length divided by the volume, area, or length 
respectively. The dimensions of electricity will be repre- 
sented by [9], and for its amount the symbol 9 will be used. 
For the volume density we will use e, for areal density e' , 
for linear density e". If the distribution may be considered 
to be continuous, we may take the limits and find the 
density at a point. 

(2) Magnetism. Considering magnetism to be a quan- 
tity, we will use for the unit of measurement the maxwell, 
connected with the absolute units by the equation 

1 maxwell = 3-10 10 electrostatic units 

= 1 electromagnetic unit. 

Sometimes 10 8 maxwells is called a weber. The symbol for 
magnetism will be $, the dimensions [$], the densities 
m, m', m". 

(3) Action. This quantity is much used in physics, the 
principle of least action being one of the most important 
fundamental bases of modern physics. The dimensions 
of action are [93>], the symbol we shall use is A, and the 
unit might be a quantum, but for practical purposes a 
joule-second is used. In the case of a moving particle the 
action at any point depends upon the path by which the 
particle has reached the point, so that as a function of the 
points of space it has at each point an infinity of values. 
A function which has but a single value at a point will be 
called monodromic, but if it has more than one value it will 
be called polydromic. The action is therefore a polydromic 
function. We not only have action in the motion of par- 
ticles but we find it as a necessary function of a momentum 
field, or of an electromagnetic field. 

(4) Energy. The unit of energy is the erg or the joule 



SCALAR FIELDS 15 

= 10 7 ergs. Its dimensions are [G^T 7-1 ], its symbol will 
beW. 

(5) Activity. This should not be confused with action. 
It is measured in watts, symbol J, dimensions [Q$T~ 2 ]. 

(6) Energy-density. The symbol will be U, dimensions 

(7) Activity-density. The symbol will be Q, dimensions 

pi- 3 r 2 ]. 

(8) Mass. The symbol is M, dimensions [0$77r 2 ]. 
The unit of mass is the gram. A distribution of mass is 
usually called a distribution of matter. 

(9) Density of mass. The symbol will be c, dimensions 

(10) Potential of electricity. Symbol V, dimensions 

(11) Potential of magnetism. Symbol N, dimensions 
[02 7 - 1 ]. 

(12) Potential of gravity. Symbol P, dimensions [G^T 7-1 ]. 

2. Levels. Points at which the function has the same 
value, are said to define a level surface of the function. It 
may have one or more sheets. Such surfaces are usually 
named by the use of the prefixes iso and equi. For instance, 
the surfaces in a cloud, which have all points at the same 
temperature, are called isothermal surfaces; surfaces which 
have points at the same pressure are called isobaric surfaces; 
surfaces of equal density are isopycnic surfaces; those of 
equal specific volume (reciprocal of the density) are the iso- 
steric surfaces; those of equal humidity are isohydric surfaces. 
Likewise for gravity, electricity, and magnetism we have 
equipotential surfaces. 

3. Lamellae. Surfaces are frequently considered for 
which we have unit difference between the values of the 
function for the successive surfaces. These surfaces and 



16 VECTOR CALCULUS 

the space between them constitute a succession of unit 
lamellae. 

If we follow a line from a point A to a point B, the number 
of unit lamellae traversed will give the difference between 
the two values of the function at the points A and B. 
If this is divided by the length of the path we shall have the 
mean rate of change of the function along the path. If 
the path is straight and the unit determining the lamellae 
is made to decrease indefinitely, the limit of this quotient 
at any point is called the derivative of the function at 
that point in the given direction. The derivative is ap- 
proximately the number of unit lamellae traversed in a 
unit distance, if they are close together. 

4. Geometric Properties. Monodromic levels cannot in- 
tersect each other, though any one may intersect itself. 
Any one or all of the levels may have nodal lines, conical 
points, pinch-points, and the other peculiarities of geo- 
metric surfaces. These singularities usually depend upon 
the singularities of the congruence of normals to the 
surface. 

In the case of functions of two variables, the scalar levels 
will be curves on the surface over which the two variables 
are defined. Their singularities may be any that can 
occur in curves on surfaces. 

5. Gradient. The equation of a level surface is found 
by setting the function equal to a constant. If, for in- 
stance, the point is located by the coordinates x, y, z 
and the function is f(x, y, z), then the equation of any 
level is 

u = /(*> V> z ) = C. 

If we pass to a neighboring point on the same surface 
we have 

du = f{x + dx, y -f- dy, z + dz) — f{x, y, z) = 0. 

We may usually find functions df/dx, bf\a\ df/dz, 



SCALAR FIELDS 17 

functions independent of dx, dy, dz, such that 

du — dfjdx • dx + df/dy • dy + df/dz • dz. 

Now the vector from the first point to the second has 
as the lengths of its projections on the axes: dx, dy, dz; and 
if we define a vector whose projections are dfjdx, df/dy, 
df/dz, which we will call the Gradient of f, then the con- 
dition du = is the condition that the gradient of / shall be 
perpendicular to the differential on the surface. Hence, 
if we represent the gradient of / by v/, and the differential 
change from one point to the other by dp, we see that dp 
is any infinitesimal tangent on the surface and v/ is along the 
normal to the surface. It is easy to see that if we differen- 
tiate u in a direction not tangent to a level surface of u we 
shall have 

du = df/dx-dx + df/dy •<&,+ df/dz -dz = dC. 

If the length of the differential path is ds then we shall have* 
du/ds = projection of^fon the unit vector in the direction of dp. 
The length of the vector v/ is sometimes called the gradient 
rather than the vector itself. Sometimes the negative of 
the expression used here is called the gradient. 

When the three partial derivatives of / vanish for the 
same point, the intensity of the gradient, measured by its 
length, is zero, and the direction becomes indeterminate 
from the first differentials. At such points there are singu- 
larities of the function. At points where the function 
becomes infinite, the gradient becomes indeterminate and 
such points are also singular points. 

6. Potentials. The three components of a vector at a 
point may be the three partial derivatives of the same 
function as to the coordinates, in which case the vector 
may be looked upon as the gradient of the integral func- 

* Since dxjds, dyjds, dzjds are the direction-cosines of dp. 



18 VECTOR CALCULUS 

tion, which is called a potential junction, or sometimes a 
force function. For instance, if the components of the 
velocity satisfy the proper conditions, the velocity is the 
gradient of a velocity 'potential. These conditions will be 
discussed later, and the vector will be freed from dependence 
upon any axes. 

7. Relative Derivatives. In case there are two scalar 
functions at a point, we may have use for the concept of 
the derivative of one with respect to the other. This is 
defined to be the quotient of the intensity of the gradient of 
the first by that of the second, multiplied by the cosine 
of their included angle. If the unit lamellae are constructed, 
it is easy to see from the definition that the relative deriva- 
tive of the first as to the second will be the limit of the 
average or mean of the number of unit sheets of the first 
traversed from one point to another, along the normal of the 
second divided by the number of unit sheets of the second 
traversed at the same time. For instance, if we draw the 
isobars for a given region of the United States and the 
simultaneous isotherms, then in passing from a point A 
to a point B if we traverse 24 isobaric unit sheets and 10 
isothermal unit sheets, the average is 2.4 isobars per 
isotherm. ^ 

8. Unit-Tubes. If there are two scalar functions in the 
field, and the unit lamellae are drawn, the unit sheets will 
usually intersect so as to divide the space under considera- 
tion into tubes whose cross-section will be a curvilinear 
parallelogram. Since the area of such parallelogram is 
approximately 

dsids2 esc 0, 
where dsi is the distance from a unit sheet of the function u 
to the next unit sheet, and ds 2 the corresponding distance 
for the function v, while 6 is the angle between the surfaces; 
and since we have, Tyu being the intensity of the gradient 



SCALAR FIELDS 19 

of u, and T^/v the intensity of the gradient of v, 

dsi - 1/TVu, ds 2 = 1/Tw 

the area of the parallelogram will be l/(TyuTvv sin 6). 
Consequently if we count the parallelograms in any plane 




Fig. 1. 

cross-section of the two sets of level surfaces, this number 
is an approximate value of the expression 

T^uT^Jv sin 6 X area parallelogram 

when summed over the plane cross-section. That is to 
say, the number of these tubes which stand perpendicular 
to the plane cross-section is the approximate integral of the 
expression T^uT^v sin 6 over the area of the cross-section. 
These tubes are called unit tubes for the same reason that 
the lamellae are called unit lamellae. 

In counting the tubes it must be noticed whether the 
successive surfaces crossed correspond to an increasing or 
to a decreasing value of u or of v. It is also clear that 
when sin 6 is everywhere the integral must be zero. In 
such case the three Jacobians 

d(u, v)/d(y, z), d(u, v)/d(z, x), d{u, v)/d(x, y) 



20 VECTOR CALCULUS 

are each equal zero, and this is the^condition that u is a 
function of v. In case the plane of cross-section is the 
x, y plane, the first two expressions vanish anyhow, since 
u, v are functions of x, y only. 

It is clear if we take the levels of one of the functions, 
say u, as the upper and lower parts of the boundary of the 
cross-section, that in passing from one of the other sides 
of the boundary along each level of u the number of unit 
tubes we encounter from that side of the boundary to the 
opposite side is the excess of the value of v on the second 
side over that on the first side. If then we count the dif- 
ferent tubes in the successive lamellae of u between the 
two sides of the cross-section we shall have the total excess 
of those on the second side over those on the first side. 
That is to say, the number of unit tubes or the integral 
over the area bounded by level 1 and level 2 of u, and any 
other two lines which cross these two levels so as to produce 
a simple area between, is the excess of the sum between 
the two levels of the values of v on one side over the same 
sum between the two levels of u on the other side. These 
graphical solutions are used in Meteorology. 

This gives the excess of the integral J vdu along the 

second line between the two levels of u, over the same in- 
tegral along the first line. It represents the increase of this 
integral in a change of path from one line to the other. For 
instance if the integral is energy, the number of tubes is 
the amount of energy stored or released in the passage from 
one line to the other, as in a cyclone. The number of tubes 

for any closed path is the approximate integral I rdu 

around the path. , 



SCALAR FIELDS 21 

EXERCISES. 

1. If the density varies as the distance from a given axis, what are 
the isopycnic surfaces? 

2. A rotating fluid mass is in equilibrium under the force of gravity, 
the hydrostatic pressure, and the centrifugal force. What are the 
levels? Show that the field of force is conservative. 

3. The isobaric surfaces are parallel planes, and the isopycnic 
surfaces are parallel planes at an angle of 10° with the isobaric planes. 
What is the rate of change of pressure per unit rate of change of density 
along a line at 45° with the isobaric planes? 

4. If the pressure can be stated as a function of the density, what 
conditions are necessary? Are they sufficient? What is the interpreta- 
tion with regard to the levels? 

5. Three scalar functions have a functional relation if their Jacobian 
vanishes. What does this mean with regard to their respective levels? 

6. If the isothermal surfaces are spheres with center at the earth's 
center, the temperature sheets for decrease of one degree being 166.66 
feet apart, and if the isobaric levels are similar spheres, the pressure 
being given by 

log B = log B, - 0.0000177 (a - z ), 

where B is the pressure at z feet above the surface of the earth, what 
is the relative derivative of the temperature as to the pressure, and the 
pressure as to the temperature? 

7. To find the maximum of u(x, y, z) we set du = 0. If there is also 
a condition to be fulfilled, v(x, y, z) = 0, then dv = also. 

These two equations in dx, dy, dz must be satisfied for all compatible 
values of dx, dy, dz, and we must therefore have 

du du du _ _ dy # dv dv_ 
dx' dy' dz' ~ dx' dy' dz } 

which is equivalent to the single vector equation 

Vw = wyv. 

What does this mean in terms of the levels : ; The unit tubes? 

If there is also another equation of condition l(x, y, z) =0 then also 
dt = and the Jacobian of the three functions u, v, t must equal zero. 
Interpret. 

8. On the line of intersection of two levels of two different functions 
the values of both functions remain constant. If we differentiate a 
third function along the locus in question, the differential vanishing 
everywhere, what is the significance? 



22 



VECTOR CALCULUS 



9. If a field of force has a potential, then a fluid, subject to the force 
and such that its pressure is a function of the density and the tempera- 
ture, will have the equipotential levels for isobaric levels also. The 
density will be the derivative of the pressure relative to the potential. 
Show therefore that equilibrium is not possible unless the isothermals 
are also the levels of force and of pressure. 

[p = p(c, T), and vp = cvv = PcVc + prvT. 

If then vc = 0, cvv = prVT.] 

10. If the full lines below represent the profiles of isobaric sheets, and 
the dotted lines the profiles of isosteric sheets, count the unit tubes 
between the two verticals, and explain what the number means. If 
they were equipotentials of gravity and isopycnic surfaces, what would 
the number of unit tubes mean? 




Fig. 2. 

11. If u = y — 12x 3 and v = y + x 2 + \x, find Vw and w and 
TvuTw -sin 6, and integrate the latter over the area between x = f 
x = 1, y = 0, y = 12. Draw the lines. 

12. If u = ax + by + cz and v = x 2 -f- if + z 2 , find vw and vv and 
TyuTvvsm 6 and integrate the latter expression over the surface of a 
cylinder whose axis is in the direction of the z axis. Find the deriva- 
tive of each relative to the other. 



CHAPTER III 

VECTOR FIELDS 

1. Hypercomplex Quantity. In the measurement of 
quantity the first and most natural invention of the mind 
was the ordinary system of integers. Following this came 
the invention of fractions, then of irrational numbers. 
With these the necessary list of numbers for mere measure- 
ment of similar quantities is closed, up to the present time. 
Whether it will be necessary to invent a further extension 
of number along this line remains for the future to show. 

In the attempt to solve equations involving ordinary 
numbers, it became necessary to invent negative numbers 
and imaginary numbers. These were known and used as 
fictitious numbers before it was noticed that quantities 
also are of a negative or an "imaginary" character. We 
find instances everywhere. In debit and credit, for ex- 
ample, we have quantity which may be looked upon as of 
two different kinds, like iron and time, but the most logical 
conception is to classify debits and credits together in the 
single class balance. One's balance is what he is worth 
when the debits and credits have been compared. If the 
preponderance is on the side of debit we consider the balance 
negative, if on the side of credit we consider the balance 
positive. Likewise, we may consider motion in each direc- 
tion of the compass as in a class by itself, never using any 
conception of measurement save the purely numerical one 
of comparing things which are exactly of the same kind 
together. But it is more logical, and certainly more general, 
to consider motions in all directions of the compass and 
of any distances as all belonging to a single class of quantity. 

23 



24 VECTOR CALCULUS 

In that case the comparison of the different motions leads 
us to the notion of complex numbers. When Wessel made 
his study of the vectors in a plane he was studying the 
hypernumbers we usually call "the complex field." The 
hypernumbers had been studied in themselves before, but 
were looked upon (rightly) as being creations of the mind 
and (in that sense correctly) as having no existence in what 
might be called the real world. However, their deduction 
from the vectors in a plane showed that they were present 
as relations of quantities which could be considered as alike. 
Again when Steinmetz made use of them in the study of 
the relations of alternating currents and electromotive 
forces, it became evident that the so-called power current 
and wattless current could be regarded as parts of a single 
complex current, and similarly for the electromotive forces. 
The laws of Ohm and Kirchoff could then be generalized so 
as to be true for the new complex quantities. In this brief 
history we find an example of the interaction of the develop- 
ments of mathematics. The inventions of mathematics 
find instances in natural phenomena, and in some cases 
furnish new conceptions by which natural phenomena can 
be regarded as containing elements that would ordinarily 
be completely overlooked. 

In space of three (or more) dimensions, the vectors 
issuing from a point in all directions and of all lengths 
furnish quantities which may be considered to be all of 
the same kind, on one basis of classification. Therefore, 
they will define certain ratios or relations which may be 
called hypernumbers. This is the class of hypernumbers 
we are particularly concerned with, though we shall occa- 
sionally notice others. Further, any kind of quantity 
which can be represented completely for certain purposes 
by vectors issuing from a point we will call vector quantity. 



VECTOR FIELDS 25 

Such quantities, for instance, are motions, velocities, 
accelerations, at least in the Newtonian mechanics, forces, 
momenta, and many others. The object of VECTOR CAL- 
CULUS is to study these hypernumbers in relation to their 
corresponding quantities, and to derive an algebra capable 
of handling them. 

We do not consider a vector as a mere triplex of ordinary numbers. 
Indeed, we shall consider two vectors to be identical when they 
represent or can represent the same quantity, even though one is ex- 
pressed by a certain triplex, as ordinary Cartesian coordinates, and the 
other by another triplex, as polar coordinates. The numerical method 
of defining the vector will be considered as incidental. 

2. Notation. We shall represent vectors for the most 
part by Greek small letters. Occasionally, however, as 
in Electricity, it will be more convenient to use the standard 
symbols, which are generally Gothic type. As indicated 
on page 12 there is a great variety of notation, and only 
one principle seems to be used by most writers, namely 
that of using heavy type for vectors, whatever the style of 
type. In case the vector is from the origin to the point 
(x, y, z) it may be indicated by 

Px, y, z> 

while for the same point given by polar coordinates r, <p, 6 
we may use 

Pr, <p, 6) 

In case a vector is given by its components as X, Y, Z we 
will indicate it by 

?x, y, z 

3. Equivalence. All vectors which have the same direc- 
tion and same length will be considered to be equivalent. 
Such vectors are sometimes called free vectors. The term 
vector will be used throughout this book, however, with no 
other meaning. 



2G VECTOR CALCULUS 

In case vectors are equivalent only when they lie on the 
same line, and have the same direction and length, they 
will be called glissants. A force applied to a rigid body 
must be considered to be a glissant, not a vector. In 
case vectors are equivalent only when they start at the 
same point and coincide, they will be called radials. The 
resultant moment of a system of glissants with respect to a 
point A is a radial from A. 

The equivalence of two vectors 

a = 

implies the existence of equalities infinite in number, for 
their projections on any other lines will then be equal. The 
infinite set of equalities, however, is reducible in an infinity 
of ways to three independent equalities. For instance, we 
may write either 
a x = ft., ay = fi y , a 2 = 13 z , or a r = B r , a <p = ^ lf> ,a lf! = /?„. 

The equivalence of two glissants implies sets of equalities 
reducible in every case to five independent equalities. The 
equivalence of two radials reduces to sets of six equalities. 

4. Vector Fields. Closely allied to the notion of radial 

is that of vector field. A vector field is a system of vectors 

each associated with a point of space, or a point of a surface, 

or a point of a line or curve. The vector is a function of 

the position of the point which is itself usually given by a 

vector, as p. The vector function may be monodromic or 

polydromic. We will consider some of the usual vector 

fields. 

EXAMPLES 

(1) Radius Vector, p [L]. This will usually be indicated 
by p. In case it is a function of a single parameter, as t, 
the points defined will lie on a curve;* in case it is a function 

* We are discussing mainly ordinary functions, not the "pathologic 
type." 



VECTOR FIELDS 27 

of two parameters, u, v, the points defined will lie on a 
surface. The term vector was first introduced by Hamilton 
in this sense. When we say that the field is p, we mean 
that at the point whose vector is p measured from the fixed 
origin, there is a field of velocity, or force, or other quantity, 
whose value at the point is p. 

(2) Velocity, a [XT 7-1 ]. Usually we will designate veloc- 
ity by c. In the case of a moving gas or cloud, each particle 
has at each point of its path a definite velocity, so that we 
can describe the entire configuration of the moving mass at 
any instant by stating what function a is of p, that is, for 
the point at the end of the radius vector p assign the velocity 
vector. The path of a moving particle will be called a 
trajectory. At each point of the path the velocity a is a 
tangent of the trajectory. 

If we lay off from a fixed point the vectors a which corre- 
spond to a given trajectory, their terminal points will 
lie on a locus called by Hamilton the hodograph of the 
trajectory. For instance, the hodographs of the orbits of 
the planets are circles, to a first approximation. If we 
multiply a by dt, which gives it the dimensions of length, 
namely an infinitesimal length along the tangent of the 
trajectory, the differential equation of the trajectory 
becomes 

dp = adt. 

The integral of this in terms of t gives the equation of the 
trajectory. 

(3) Acceleration. t[LT~ 2 ]. An acceleration field is simi- 
lar to a velocity field except in dimensions. The accelera- 
tion is the rate of change of the vector velocity at a point, 
consequently, if a point describes the hodograph of a trajec- 
tory so that its radius vector at a given time is the velocity 
in the trajectory at that time, the acceleration will be a 

3 



L\S VECTOR CALCULUS 

tangent to the hodograph, and its length will be the velocity 
of the moving point in the hodograph. We will use r to 
indicate acceleration. 

(4) Momentum Density. T [$QL~ 4 ]. This is a vector 
function of points in space and of some number which can 
be attached to the point, called density. In the case of a 
moving cloud, for instance, each point of the cloud will have 
a velocity and a density. The product of these two factors 
will be a vector whose direction is that of the velocity and 
whose length is the product of the length of the velocity 
vector and the density. However, momentum density 
may exist without matter and without motion. In electro- 
dynamic fields, such as could exist in the very simple case 
of a single point charge of electricity and a single magnet 
pole at a point, we also have at every point of space a 
momentum density vector. This may be ascribed to the 
hypothetical motion of a hypothetical ether, but the essen- 
tial feature is the existence of the field. If we calculate the 
integral of the projection of the momentum density on the 
tangent to a given curve from a point A to a point B, the 
value of the integral is the action of an infinitesimal volume, 
an action density, along that path from A to B. The 
integration over a given volume would give the total 
action for all the particles over their various paths. This 
would be a minimum for the paths actually described as 
compared with possible paths. Specific momentum is 
momentum density of a moving mass. 

(5) Momentum. Y [TOL -1 ]. The volume integral of 
momentum density or specific momentum is momentum. 
Action is the line-integral of momentum. 

(6) Force Density. F [^QL^T- 1 ]. If a field of momen- 
tum density is varying in time then at each point there is a 
vector which may be called force-density, the time derivative 



VECTOR FIELDS 29 

of the momentum density. Such cases occur in fields due 
to moving electrons or in the action of a field of electric 
intensity upon electric density, or magnetic intensity on 
magnetic density. 

(7) Force. X [mL- 1 ? 7 - 1 ]. The unit of force has re- 
ceived a name, dyne. It is the volume integral of force 
density. The time integral of a field of force is momentum. 
In a stationary field of force the line integral of the field 
for a given path is the difference in energy between the 
points at the ends of the path, or what is commonly called 
work. In case the field is conservative the integral has the 
same value for all paths (which at least avoid certain 
singular points), and depends only on the end points, 
This takes place when the field is a gradient field of a force- 
function, or a potential function. If we project the force 
upon the velocity at each point where both fields exist, 
the time integral of the scalar quantity which is the product 
of the intensity of the force, the intensity of the velocity 
and the cosine of the angle between them, is the activity at 
the point. 

(8) Flux Density. 12 [UT~ 1 }. In the case of the flow of 
an entity through a surface the limiting value of the amount 
that flows normally across an infinitesimal area is a vector 
whose direction is that of the outward normal of the surface, 
and whose intensity is the limit. In the case of a flow not 
normal to the surface across which the flux is to be de- 
termined, we nevertheless define the flux density as above. 
The flux across any surface becomes then the surface 
integral of the projection of the flux density on the normal 
of the surface across which the flux is to be measured. 

Flux density is an example of a vector which depends 
upon an area, and is sometimes called a bivector. The 
notion of two vectors involved in the term bivector may 



30 VECTOR CALCULUS 

be avoided by the term cycle, or the term feuille. It is 
also called an axial vector, in opposition to the ordinary 
vectors, called polar vectors. The term axial is applicable 
in the sense that it is the axis or normal of a portion of a 
surface. The portion (feuille, cycle) of the surface is 
traversed in the positive direction in going around its 
boundary, that is, with the surface on the left-hand. If 
the direction of the axial vector is reversed, we also traverse 
the area attached in the reverse direction, so that in this 
sense the axial vector may be regarded as invariant for 
such change while the polar vector would not be invariant. 
The distinction is not of much importance. The important 
idea is that of areal integration for the flux density or any 
other so-called axial vector, while the polar vector is sub- 
ject only to linear integration. We meet the distinction 
in the difference below between the induction vectors and 
the intensity vectors. 

(9) Energy Density Current. R [TOL -2 ? 7 - 2 ]. When an 
energy density has the idea of velocity attached to it, it 
becomes a vector with the given dimensions. In such 
case we consider it as of the nature of a flux density. 

(10) Energy Current. 2 [$QT~ 2 ]. If a vector of energy 
density current is multiplied by an area we arrive at an 
energy current. 

(11) Electric Density Current. J [SL^T- 1 ]. A number 
of moving electrons will determine an average density 
per square centimeter across the line of flow, and the product 
of this into a velocity will give an electric density current. 
To this must also be added the time rate of change of 
electric induction, which is of the same dimensions, and 
counts as an electric density current. 

(12) Electric Current. C [97 1-1 ]. The unit is the ampere 
= 3-10 9 e.s. units = 10 _1 e.m. units. This is the product 
of an electric density current by an area. 






VECTOR FIELDS 31 



(13) Magnetic Density Current. G [$Ir 2 T- 1 }. Though 
there is usually no meaning to a moving mass of magnetism, 
nevertheless, the time rate of change of magnetic induction 
must be considered to be a current, similar to electric 
current density. 

(14) Magnetic Current. K [^T' 1 ]. The unit is the 
heavy side = 1 e.m. unit = 3 • 10 10 e.s. units. In the phenom- 
ena of magnetic leakage we have a real example of what may 
be called magnetic current. 

Both electric current and magnetic current may also be 
scalars. For instance, if the corresponding flux densities 
are integrated over a given surface the resulting scalar 
values would give the rate at which the electricity or the 
magnetism is passing through the surface per second. In 
such case the symbols should be changed to corresponding 
Roman capitals. 

(15) Electric Intensity. E fMr 1 ! 1 " 1 ]. When an electric 
charge is present in any portion of space, there is at each 
point of space a vector of a field called the field of electric 
intensity. The same situation happens when lines of 
magnetic induction are moving through space with a given 
velocity. The electric intensity will be perpendicular to 
both the line of magnetic induction and to the velocity it 
has, and equal to the product of their intensities by the 
sine of their angle. 

The electric intensity is of the nature of a polar vector 
and its flux, or surface integral over any surface has no 
meaning. Its line integral along any given path, however, 
is called the difference of voltage between the two points at 
the ends of the path, for that given path. The unit of 
voltage is the volt = J • 10~ 2 e.s. units = 10 8 e.m. units. 
The symbol for voltage is V [$T~ 1 ]. Its dimensions are 
the same as for scalar electric potential, or magnetic current. 



32 VECTOR CALCULUS 

(16) Electric Induction. D [QL~ 2 ]. The unit is the line 
= 3-10 9 e.s. units — 10 -1 e.m. units. This vector usually 
has the same direction as electric intensity, but in non- 
isotropic media, such as crystals, the directions do not agree. 
It is a linear function of the intensity, however, ordinarily 

indicated by 

D = k(E) 

where k is the symbol for a linear operator which converts 
vectors into vectors, called here the permittivity, [0^> -1 Z _1 T], 
measurable in farads per centimeter. In isotropic media 
k is a mere numerical multiplier with the proper dimensions, 
which are essential to the formulae, and should not be 
neglected even when k = 1. The flux is measured in 
coulombs. 

(17) Magnetic Intensity. H [eL" 1 ? 7 " 1 ]. The field due to 
the poles of permanent magnets, or to a direct current 
traversing a wire, is a field of magnetic intensity. In case 
we have moving lines of electric induction, there is a field of 
magnetic intensity. It is of a polar character, and its 
flux through a surface has no meaning. The line integral 
between two points, however, is called the gilbertage between 
the points along the given path, the unit being the gilbert 
= 1 e.m. unit = 3 • 10 10 e.s. units. The symbol is N [GT- 1 ]' 
Its dimensions are the same as those of scalar magnetic 
potential, or electric current. 

(18) Magnetic Induction. B [$L~ 2 ]. The unit is the 
gauss = 1 e.m. unit = 3 • 10 10 e.s. units. The direction is 
usually the same as that of the intensity, but in any case is 
given by a linear vector operator so that we have 

B-m(H) 

where \x is the inductivity, [^>0 -1 Z _1 T], measurable in henrys 
per centimeter. The flux is measured in maxwells. 



VPPf 



VECTOR FIELDS 33 



(19) Vector Potential of Electric Induction. T [eZ -1 ]. A 
vector field may be related to another vector field in a 
certain manner to be described later, such that the first 
can be called the vector potential of the other. 

(20) Vector Potential of Magnetic Induction. ^ [M -1 ]. 
This is derivable from a field of magnetic induction. This 
and the preceding are line-integrable. 

(21) Hertzian Vectors. 9, <£. These are line integrals of 
the preceding two, and are of a vector nature. 

5. Vector Lines. If we start at a given point of a vector 
field and consider the vector of the field at that point to be 
the tangent to a curve passing through the point, the field 
will determine a set of curves called a congruence, since there 
will be a two-fold infinity of curves, which will at every 
point have the vector of the field as tangent. If the field 
is represented by a, a function of p, the vector to a point of 
the field, then the differential equation of these lines of 
the congruence will be 

dp = adt, 

where dt is a differential parameter. From this we can 
determine the equation of the lines of the congruence, in- 
volving an arbitrary vector, which, however, will not have 
more than two essential constants. For instance, if the 
field is given by a = p, then dp = pdt, and p = ae l , where 
a is a constant unit vector. The lines are, in this case, the 
rays emanating from the origin. 

The lines can be constructed approximately by starting 
at any given point, thence following the vector of the field 
for a small distance, from the point so reached following 
the new vector of the field a small distance, and so proceed- 
ing as far as necessary. This will trace approximately a 
vector line. Usually the curves are unique, for if the field 
is monodromic at all points, or at points in general, the 



34 VECTOR CALCULUS 

curves must be uniquely determined as there will be at any 
point but one direction to follow. Two vector lines may 
evidently be tangent at some point, but in a monodromic 
field they cannot intersect, except at points where the in- 
tensity of the field is zero, for vectors of zero intensity are 
of indeterminate direction. Such points of intersection 
are singular points of the field, and their study is of high 
importance, not only mathematically but for applications. 
In the example above the origin is evidently a singular 
point, for at the origin a = 0, and its direction is indetermi- 
nate. 

6. Vector Surfaces, Vector Tubes. In the vector field 
we may select a set of points that lie upon a given curve 
and from each point draw the vector line. All such vector 
lines will lie upon a surface called a vector surface, which in 
case the given curve is closed, forming a loop, is further 
particularized as a vector tube. It is evident that the vector 
lines are the characteristics of the differential equation 
dp = adt, which in rectangular coordinates would be 
equivalent to the equations 

dx _dy _ dz 
X ~ Y~ Z' 

In case these equations are combined so as to give a 
single exact equation, the integral will (since it must con- 
tain a single arbitrary constant) be the equation of a family 
of vector surfaces. The vector lines are the intersections 
of two such families of vector surfaces. The two families 
may be chosen of course in infinitely many different ways. 
Usually, however, as in Meteorology, those surfaces are 
chosen which have some significance. When a vector 
tube becomes infinitesimal its limit is a vector line. 

7. Isogons. If we locate the points at which a has the 



VECTOR FIELDS 35 

same direction, they determine a locus called an isogon for 
the field. For instance, we might locate on a weather map 
all the points which have the same direction of the wind. 
If isogons are constructed in any way it becomes a simple 
matter to draw the vector lines of the field. Machines for 
the use of meteorologists intended to mark the isogons 
have been invented and are in use.* As an instance con- 
sider the vector field 

a = (2x, 2y, — z). 

An isogon with the points at which a has the direction whose 
cosines are /, m, n is given by the equations 

2x : 2y : — z = I : m : n 
or 

2x = It, 2y = mt, z = — nt. 

It follows that the vector to any point of this isogon is 
given by 

p = t(l, m, n) - (0, 0, 3nt). 

That is to say, to draw the vector p to any point of the 
isogon we draw a ray from the origin in the direction given, 
then from its outer end draw a parallel to the Z direction 
backward three times the length of the Z projection of the 
segment of the ray. The points so determined will evi- 
dently lie on straight lines in the same plane as the ray and 
its projection on the XY plane, with a negative slope twice 
the positive slope of the ray. The tangents of the vector 
lines passing through the points of the isogon will then be 
parallel to the ray itself. The vector lines are drawn ap- 
proximately by drawing short segments along the isogon 
parallel to its corresponding ray, and selecting points such 
that these short segments will make continuous lines in 
*Sandstrdm: Annalen der Hydrographie und Maritimen Meteor- 
ologie (1909), no. 6, pp. 242 et.seq. Bjerknes: Dynamic Meteorology. 
See plates, p. 50. 



36 



VECTOR CALCULUS 



passing to adjacent isogons. The figure illustrates the 
method. All the vector lines are found by rotating the 
figure about the X axis 180°, and then rotating the figure 
so produced about the Z axis through all angles. 




Fig. 3. 

8. Singularities. It is evident in the example preceding 
that there are in the figure two lines which are different 
from the other vector lines, namely, the Z axis and the line 
which is in the XY plane. Corresponding to the latter 
would be an infinity of lines in the XY plane passing through 
the origin. These lines are peculiar in that the other vector 
lines are asymptotic to them, while they are themselves 
vector lines of the field. A method of studying the vector 
lines in the entire extent of the plane in which they lie was 
used by Poincare. It consists in placing a sphere tangent 



VECTOR FIELDS 37 

to the plane at the origin. Lines are then drawn from the 
center of the sphere to every point of the plane, thus giving 
two points on the sphere, one on the hemisphere next the 
plane and one diametrically opposite on the hemisphere 
away from the plane. The points at infinity in the plane 
correspond to the equator or great circle parallel to the 
plane. In this representation every algebraic curve in the 
plane gives a closed curve or cycle on the sphere. In the 
present case, the axes in the plane give two perpendicular 
great circles on the sphere, and the vector lines will be 
loops tangent to these great circles at points where they 
cross the equator. These loops will form in the four Junes 
of the sphere a system of closed curves which Poincare calls 
a topographical system. The equator evidently belongs to 
the system, being the limit of the loops as they grow nar- 
rower. The. two great circles corresponding to the axes 
also belong to the system, being the limits of the loops as 
they grow larger. If a point describes a vector line its 
projection on the sphere will describe a loop, and could 
never leave the lune in which the projection is situated. 
The points of tangency are called nodes', the points which 
represent the origin, and through which only the singular 
vector lines pass, are called fames. 

9. Singular Points. The simplest singular lines depend 
upon the singular points and these are found comparatively 
simply. The singular points occur where 

o" = or a —• oo . 

Since we may multiply the components of a by any ex- 
pressions and still have the lines of the field the same, we 
may equally suppose that the components of a are reduced 
to as low terms as possible by the exclusion of common 
factors of all of them. We will consider first the singular 



38 VECTOR CALCULUS 

points for fields in space, then those cases which have 
lines every point of which is a singular point, which will 
include the cases of plane fields, since these latter may be 
considered to represent the fields produced by moving the 
plane field parallel to itself. The classification given by 
Poincare is as follows. 

(1) Node. At a node there may be many directions 
in which vector lines leave the point. An example is a = p. 
At the origin, it is easy to see, a = 0, and it is not possible 
to start at the origin and follow any definite direction. 
In fact the vector lines are evidently the rays from the 
origin in all directions. There is no other singular point at 
a finite distance. If, however, we consider all the rays in 
any one plane, and for this plane construct the sphere of 
projection, we see that the lines correspond to great circles 
on the sphere which all pass through the origin and the 
point diametrically opposite to it. This ideal point may 
be considered to be another node, so that all the vector 
lines run from node to node, in this case. Every vector 
line which does not terminate in a node is a spiral or a cycle. 

(2) Faux. From a faux* there runs an infinity of vector 
lines which are all on one surface, and a single isolated 
vector line which intersects the surface at the faux. The 
surface is a singular surface since every vector line in it 
through the faux is a singular line. The singular surface 
is approached asymptotically by all the vector lines not 
singular. 

An example is given by 

a = (x, y, — z). 

The vector lines are to be found by drawing all equilateral 
hyperbolas in the four quadrants of the ZX plane, and then 
* Poincare uses the term col, meaning mountain pass, for which faux 
is Latin. 



/ 



VECTOR FIELDS 



39 



rotating this set of lines about the Z axis. Evidently all 
rays in the XY plane from the origin are singular lines, as 
well as the Z axis. Where fauces occur the singular lines 
through them are asymptotes for the nonsingular lines. If 







Fig. 4. 

we consider any plane through the Z axis, the system of 
equilateral hyperbolas will project onto its sphere as cycles 
tangent on the equator to the great circles which repre- 
sent the singular lines in that plane. From this point of 
view we really should consider the two rays of the Z axis as 
separate from each other, so that the upper part of the Z 
axis and the singular ray perpendicular to it, running in the 
same general direction as the other vector lines, would con- 
stitute a vector line with a discontinuity of direction, or 
with an angle. Such a vector line to which the others are 
tangent at points at infinity only is a boundary line in the 
sense that on one side we have infinitely many vector lines 
which form cycles (in the sense defined) while on the other 
sides we have vector lines which belong to different sys- 
tems of cycles. 



40 



VECTOR CALCULUS 



A simple case of this example might arise in the inward 
flow of air over a level plane, with an ascending motion 
which increased as the air approached a given vertical 
line, becoming asymptotic to this vertical line. In fact, 
a small fire in the center of a circular tent open at the bottom 
for a small distance and at the vertex, would give a motion 
to the smoke closely approximating to that described. 

A singular line from a faux runs to a node or else is a 
spiral or part of a cycle which returns to the faux. 

An example that shows both preceding types is the field 

a = (x 2 + y 2 — 1, bxy — 5, mz). 




In the X Y plane the singular points are at infinity as follows : 
A at the negative end of the X axis, and B at the positive 
end, both fauces; C at the end of the ray whose direction 
is tan -1 2, in the first quadrant, D at the end of the ray of 
direction tan -1 2 in the third quadrant; E at the end of the 



VECTOR FIELDS 41 

ray of direction tan -1 — 2 in the fourth quadrant; and F 
at the end of the ray of tan -1 — 2 in the second quadrant, 
these four being nodes. Vector lines run from E to D 
separated from the rest of the plane by an asymptotic 
division line from B to D; from C to D on the other side 
of this division line, separated from the third portion of 
the plane by an asymptotic division line from C to A ; and 
from C to F in the third portion of the plane. The figure 
shows the typical lines of the field. 

(3) Focus. At a focus the vector lines wind in asymp- 
totically, either like spirals wound towards the vertex of a 
spindle produced by rotating a curve about one of its 
tangents, one vector line passing through the focus, or 
they are like spirals wound around a cone towards the 




Fig. 6. 
vertex. As an example 

o- = (x+ y, y - x, z). 

The Z axis is a single singular line through the origin, which 
is a singular point, a focus in this case. The XY plane 
contains vector lines which are logarithmic spirals wound in 
towards the origin. The other vector lines are spirals 



42 



VECTOR CALCULUS 



wound on cones of revolution, their projections on XY 
being the logarithmic spirals. By changing z to az we 
would have different surfaces depending upon whether 

1 < a. 



a< 1 



or 



In case a spiral winds in onto a cycle, the successive 
turns approaching the cycle asymptotically, the cycle is 
called a limit cycle. In this example the line at infinity 
in the X Y plane, or the corresponding equator on its sphere, 
is a limit cycle. It is clear that the spirals on the cones 
wind outward also towards the lines at infinity as limit 
cycles. From this example it is plain that vector lines 
which are spiral may start asymptotically from a focus and 
be bounded by a limit cycle. The limit cycle thus divides 
the plane or the surface upon which they lie into two 
mutually exclusive regions. Vector lines may also start 
from a limit cycle and proceed to another limit cycle. 

As an example of vector lines of both kinds consider the 
field 




Fig. 7. 
a = ( r 2 _ 1, r 2 + lf mz)f 

where the first component is in the direction of a ray in the 
XY plane from the origin, the second perpendicular to 



VECTOR FIELDS 



43 



this in the XY plane, and the third is parallel to the Z axis. 
The vector lines in the singular plane, the XY plane, are 
spirals with the origin as a focus for one set, which wind 
around the focus negatively and have the unit circle as a 
limit cycle, while another set wind around the unit circle 
in the opposite direction, having the line at infinity as a 
limit cycle. The polar equation of the first set is r~ l — r 



An example with all the preceding kinds of singularities 
is the field 




Fig. 8. 

a = ( [r 2 - l)(r - 9)], (r 2 - 2r cos 9 - 8), mz) 

with directions for the components as in the preceding 
example. The singular points are the origin, a focus; the 
point A (r = 3, = + cos -1 §), a node; the point B (r = 3, 
6 = — cos -1 J), a faux. The line at infinity is a limit 
cycle, as well as the circle r = 1, which is also a vector 
line. The circle r = 3 is a vector line which is a cycle, 

4 



44 VECTOR CALCULUS 

starting at the faux, passing through the node and returning 
to the faux. The vector lines are of three types, the first 
being spirals that wind asymptotically around the focus, 
out to the unit circle as limit cycle; the second start at the 
node A and wind in on the unit circle as limit cycle; the 
third start at the node A and wind out to the line at in- 
finity as unit cycle. The second set dip down towards the 
faux. The exceptional vector lines are the line at infinity, 
the unit circle, both being limit cycles; the circle of radius 
3; a vector line which on the one side starts at the faux B 
winding in on the unit circle, and on the other side starts 
at the faux B winding outward to the line at infinity as 
limit cycle. The last two are asymptotic division lines of 
the regions. The figure exhibits the typical curves. 

(4) Faux-Focus. This type of singular point has passing 
through it a singular surface which contains an infinity 
of spirals having the point as focus, while an isolated vector 
line passes through the point and the surface. No other 
surfaces through the vector lines approach the point. An 
instance is the field 

a- = (x, y, — z). 

The Z axis is the isolated singular line, while the XY plane 
is the singular plane. In it there is an infinity of spirals 
with the origin as focus and the line at infinity as limit 
cycle. All other vector lines lie on the surfaces rz = const. 
These do not approach the origin. 

(5) Center. At a center there is a vector line passing 
through the singular point, and not passing through this 
singular line there is a singular surface, with a set of loops 
or cycles surrounding the center, and shrinking upon it. 
There is also a set of surfaces surrounding the isolated 
singular line like a set of sheaths, on each of which there are 
vector lines winding around helically on it with a decreasing 



VECTOR FIELDS 



45 




Fig. 9. 



pitch as they approach the singular surface, which they 
therefore approach asymptotically. As an instance we 
have the field 

a = (y, - x, z). 

The Z axis is the singular isolated vector line, the XY plane 

the singular surface, circles 

concentric to the origin the 

singular vector lines in it, and 

the other vector lines lie on 

circular cylinders about the 

Z axis, approaching the XY 

plane asymptotically. 

The method of determining 
the character of a singular 
point will be considered later 
in connection with the study 
of the linear vector operator. 
A singular point at infinity is either a node or a faux. 

10. Singular Lines. Singularities may not occur alone 
but may be distributed on lines every point of which is a 
singular point. This will evidently occur when cr = gives 
three surfaces which intersect in a single line. The dif- 
ferent types may be arrived at by considering the line of 
singularities to be straight, and the surfaces of the vector 
lines with the points of the singular line as singularities 
to be planes, -for the whole problem of the character of the 
singularities is a problem of analysis situs, and the deforma- 
tion will not change the character. The types are then as 
follows : 

(1) Line of Nodes. Every point of the singular line is a 
node. A simple example is a = (x, y, 0). The vector 
lines are all rays passing through the Z axis and parallel 
to the XY plane. 



46 VECTOR CALCULUS 

(2) Line of Fauces. There are two singular vector 
lines through each point of the singular line. As an instance 
a = (x, — y, 0). The lines through the Z axis parallel to 
the X and the Y axes are singular, all other vector lines 
lying on hyperbolic cylinders. 

(3) Line of Foci. The points of the singular line are 
approached asymptotically by spirals. As an instance 
<t = (x + y, y — x, 0). The vector lines are logarimithic 
spirals in planes parallel to the XY plane, wound around the 
Z axis which is the singular line. 

(4) Line of Centers. A simple case is a — (y, — x, 0). 
The vector lines are the Z axis and all circles with it as axis. 

11. Singularities at Infinity. The character of these is 
determined by transforming the components of a so as to 
bring the regions at infinity into the finite parts of the 
space we are considering. The asymptotic lines will then 
have in the transformed space nodes at which the lines are 
tangent to the asymptotic line. 

12. General Characters. The problem of the character 
of a vector field so far as it depends upon the vector lines 
and their singularities is of great importance. Its general 
resolution is due to Poincare. In a series of memoirs in 
the Journal des Mathematiques* he investigated the 
qualitative character of the curves which represent the 
characteristics of differential equations, particularly with 
the intention of bringing the entire set of integral curves 
into view at once. Other studies of differential equations 
usually relate to the character of the functions defined at 
single points and in their vicinities. The chief difficulty 
of the more general study is to ascertain the limit cycles. 
These with the asymptotic division lines separate the 
field into independent regions. 

* Ser. (3) 7 (1881), p. 375; ser. (3) 8 (1882), p. 251; ser. (4) 1 (1885), 
p. 167. Also Takeo Wado, Mem. Coll. Sci. Tokyo, 2 (1917) 151. 



VECTOR FIELDS 47 

The asymptotic division lines appear on meteorological 
maps as lines on the surface of the earth towards which, 
or away from which, the air is moving. They are called 
in the two cases lines of convergence, or lines of divergence, 
respectively. If a division line of this type starts at a 
node the node may be a point of convergence or a point of 
divergence. The line will then have the same character. 
The node in other fields, such as electric or magnetic or heat 
flow, is a source or a sink. If a division line starts from a 
faux, the latter is often called a neutral point. A focus may 
be also a point of convergence or point of divergence. In 
the case of a singular line consisting of foci, the singular line 
may be a line of convergence or of divergence; in the first 
case, for instance, the singular line is the core of the anti- 
cyclone, in the latter case, the core of the cyclone. 

The limit cycles which are not at infinity are division 
lines which enclose areas that remain isolated in the field. 
Such phenomena as the eye of the cyclone illustrate the oc- 
currence of limit cycles in natural phenomena. The limit 
cycle may be a line of convergence or a line of divergence, 
the air in the first case flowing into the line asymptotically 
from both inside and outside, with the focus serving as a 
source, and in the other case with conditions reversed. 

The practical handling of these problems in meteorological 
work depends usually upon the isogonal lines: the lines 
which are loci of equidirected tangents of the vector lines 
of the field. These are drawn and the infinitesimal tan- 
gents drawn across them. The filling in of the vector 
lines is then a matter of draughtsmanship. The isogonal 
lines will themselves have singularities and these will 
enable one to determine somewhat the singularities of the 
vector lines themselves. Since the unit vector in the 
direction of a is constant along an isogon it is evident that 



48 VECTOR CALCULUS 

the only change in a along an isogon is in its intensity, 
that is, a keeps the same direction, and its differential is 
therefore a multiple of a, that is, the isogons have for their 
differential equation 

da = adt. 

Consequently, when a = or a = <x> the isogon will have a 
singular point. It does not follow, however, that all the 
singular points of the isogons will appear as singular points 
such as are described above for the vector lines. When 
the differential equation of the isogons is reduced to the 
standard form 

dp = rdu 

we shall see later that r will be a linear vector function of a, 
and that a linear vector function may have zero directions, 
so that <pa — 0, without a = 0. Some of the phenomena 
that may happen are the following, from Bjerknes' Dynamic 
Meteorology and Hydrography. See his plates 42a, 426. 

1. Node of Isogons. These may be positive, in which 
case the directions of the tangents of the vector lines will 
increase (that is, the tangent will turn positively) as succes- 
sive isogons are taken in a positive rotation about the node, 
or may be negative in the reverse case. The positive node 
of the isogon will then correspond to a node, a focus, or a 
center of the vector lines. The negative node of the isogon 
will correspond to a faux of the vector lines. 

If the isogons are parallel, having, therefore, a node at 
infinity in either of their directions, the vector lines may 
have asymptotic division lines running in the same direc- 
tion, or they may have lines of inflexion parallel to the 
isogons. 

2. Center of Isogons. When the isogons are cycles they 
may correspond to very complicated forms of the vector 



VECTOR FIELDS 40 

lines. Several of these are to be found in a paper by Sand- 

strom, Annalen der Hydrographie und maritimen Meteor- 

ologie, vol. 37 (1909), p. 242, Uber die Bewegung der 

Flussigkeiten. 

EXERCISES* 

* To be solved graphically as far as possible. 

1. A translation field is given by a- = (at, bt, ct), what are the vector 
lines, the isogons, and the singularities? 

2. A rotation field is given by a = (mz — ny, nx — Iz, ly — mx), 
what are the isogons, singularities, and vector lines? 

3. A field of deformation proportional to the distance in one direction 
is given by a = {ax, 0, 0). Determine the field. 

4. A general field of linear deformation is given by 

o- = (ax + by + cz, fx + gy + hz, kx + ly + tnz) . 
determine the various kinds of fields this may represent according to 
the different possible cases. 

5. Consider the quadratic field* 

a = (x 2 — y 2 — z 2 , 2xy, 2xz). 

6. Consider the quadratic field a = (xy — xz, yz — yx, zx — zy). 

7. What are the lines of flow when the motion is stationary in a 
rotating fluid contained in a cylindrical vessel with vertical axis of 
rotation? 

8. Consider the various fields a = (ay -\- x, y — ax, b) for different 
values of a, which is the tangent of the angle between the curves and 
their polar radii. What happens in the successive diagrams to the 
isogons, to the curves? 

9. Consider the various fieldsf a = (l,f(r — a), b) where r is the 
polar radius in the XY plane, a is constant, and / takes the various 
forms 

f(x) = x, x 2 , x 3 , x 112 , x 113 , x~ l , x~ 2 , e x , log x, sin x, tan x. 

10. Consider the forms a = (1, f(air sin r), b) where 

j(x) = sin x, cos x, tan x. 

11. In various electrical texts, such as Maxwell, Electricity and 
Magnetism, and others, there will be found plates showing the lines of 
various fields. Discuss these. Also, the meteorological maps in 
Bjerknes' Dynamic Meteorology, referred to fibove. 

* See Hitchcock, Proc. Amer. Acad. Arts and Sci., 12 (1917), No. 7, 
pp. 372-454. 

f See Sandstrom cited above. 



50 VECTOR CALCULUS 

12. In a funnel-shaped, vortex of a water-spout the spout may be 
considered to be made up of twisted funnels, one inside another, the 
space between the surfaces being a vortex tube. In the Cottage City 
water-spout, Aug. 19, 1896, the equation of the outside funnel may be 
taken to be 

(z 2 + y*)z = 3600. 

In this x, y are measured horizontally in meters from the axis of the 
tubes, and z is measured vertically downwards from the cloud base, 
which is 1100 meters above the ground. The inner surfaces have the 
same equation save that instead of 3600 on the right we have 
3600/(1. 60 10) 2n ; that is, at any level, the radius of a surface bounding 
a tube is found from the preceding radius at the same level by dividing 
by the number whose logarithm (base 10) is 0.20546. From meteoro- 
logical theory the velocity of the wind on any surface is given by 

<r = (Cr, Crz, - 2Cz) 

where the first component is the horizontal radial component, the 
second is the tangential, and the third is the vertical component. C 
varies for the different surfaces, and is found by multiplying the value 
for the outside surface by the square of the number 1.6010. In Bige- 
low's Atmospheric Radiation, etc., p. 200 et seq., is to be found a set of 
tables for the various values from these data for different levels. Char- 
acterize the vortex field of the water-spout. 

13. For a dumb-bell-shaped water-spout, likewise, the funnels have 

the equation 

(x 2 + y 2 ) sin az — const/A 

where A varies from surface to surface just as C in the preceding 
problem. The velocity is given by 

o- = (— Aar cos az, Aar sin az, 2A sin az), 

the directions being horizontal radial, tangential and vertical. For 
the St. Louis tornado, May 27, 1896, the following data are given. 
Cloud base 1200 meters above the ground, divided into 121 parts 
called degrees, the ground thus being at 60°, and az being in degrees. 
The values of A are for the successive funnels 

0.1573, 0.4052, 1.0437, 2.6883, 6.9247, 17.837. 

Characterize the vector Ikies of this vortex field. 

14. In the treatise on The Sun's Radiation, Bigelow gives the follow- 
ing data for a funnel-shaped vortex 

r 2 z = 6400000/C 




W=wind--from9f 



PLATE I 




PLATE II 



VECTOR FIELDS 51 

at 500 kilometers z = 500, r = 60474, 26287, 11513, 5023, 2192, 956. 

a (Km/sec) = (Cr, Crz, - 2Cz). 
Calculate for 

z = 0, 500, 1000, 2000, 5000, 10000, 20000, 30000, 40000, 50000. 

The results of the calculations give a vortex field agreeing with Hale's 
observations. 

The vector lines in the last three problems lie on the funnel surfaces, 
being traced out in fact by a radius rotating about the axis of the vortex, 
and advancing along the axis according to the law 

2d = - z + C for the funnel, 
20 = az + C for the dumb-bell. 

15. Study the lines on the plates, which represent on the first plate 
the isogons for wind velocities, on the second plate the corresponding 
characteristic lines of wind flow. The date was evening of Jan. 9, 1908. 
European and American systems of numbering directions are shown in 
the margin of plate 1. See Sandstrom's paper cited above. 

13. Congruences. We still have to consider the relations 
of the various vector lines to each other, noticing that the 
vector lines constitute geometrically a congruence, that is, 
a two-parameter system of curves in space. The con- 
sideration of these matters, however, will have to be post- 
poned to a later chapter. 



CHAPTER IV 

ADDITION OF VECTORS 

1 . Sum of Vectors. Geometrically, the sum of two or 
more vectors is found by choosing any one of them as the 
first, from the terminal point of the first constructing the 
second (any other), from the terminal point of this con- 
structing the third (any of those left) and so proceeding 
till all have been successively joined to form a polygon in 
space with the exception of a final side. If now this last 
side is constructed by drawing a vector from the initial 
point of the first to the terminal point of the last, the vector 
so drawn is called the sum of the several vectors. In 
case the polygon is already closed the sum is a zero vector. 
When the sum of two vectors is zero they are said to be 
opposite, and subtraction of a vector consists in adding its 
opposite. 

It is evident from the definition that we presuppose a space in which 
the operations can be effectively carried out. For instance, if the space 
were curved like a sphere, and the sum of two vectors is found, it would 
evidently be different according to which is chosen as the first. The 
study of vector addition in such higher spaces has, however, been con- 
sidered. Encyclopedic des sciences mathematiques, Tome IV, Vol. 2. 

2. Algebraic Sum. In order to define the sum without 
reference to space, it is necessary to consider the hyper- 
numbers that are the algebraic representatives of the 
geometric vectors. We must indeed start with a given 
set of hypernumbers, 

which are the basis of the system of hypernumbers we in- 
tend to study. These are sometimes called imaginaries, 
because they are analogous to V— 1. In the case of three- 

52 



ADDITION OF VECTORS 53 

dimensional space there are three such hypernumbers in the 
basis. We combine in thought a numerical value with 
each of these, the field or domain from which these numeri- 
cal values are chosen being of great importance. For in- 
stance, we may limit our numbers to the domain of integers, 
the domain of rationals, the domain of reals, or to other 
more complicated domains, such as certain algebraic fields. 
We then consider all the multiplexes we can form by put- 
ting together into a single entity several of the hypernum- 
bers just formed, as for instance, we would have in three- 
dimensional space such a compound as 

p = («1, 7/e 2 , Z€ 3 ). 

Since we are now using the base hypernumbers e it is no 
longer necessary to use the parentheses nor to pay attention 
to the order of the terms. We drop the use of the comma, 
however, and substitute the + sign, so that we would now 
write 

p = X€i + 2/€ 2 + 2€ 3 . 

We may now easily define the algebraic sum of several 
hypernumbers corresponding to vectors by the formula 

Pi = Xi€i + y { €2 + Zi*z, [ i = 1, 2, • • • m, 

]T Pi = 2£i€i + 2^€ 2 + 2Zi€ 3 . 

i = 1 

This definition of course includes subtraction as a special 
case. 

It is clear from this definition that to correspond to the 
geometric definition, it is necessary that the units e corre- 
spond to three chosen unit vectors of the space under con- 
sideration. They need not be orthogonal, however. The 
coefficients of the e are then the oblique or rectangular 
coordinates of the point which terminates the vector if it 
starts at the origin. 



54 VECTOR CALCULUS 

3. Change of Basis. We may define all the hyper- 
numbers of the system in terms of a new set linearly related 
to the original set. For instance, if we write 

€1 = duOti + ai2«2 + Ol3«3, 
€ 2 = CiziOLl + a22«2 "T" 023«3> 

€3 = a 3 iai + a 32 a 2 + «33«3, 
then p becomes 

P = (a n x + a n y + a n z)ai 

+ (a u x + a 22 y + a 32 z)a 2 + (a n x -f a 2z y + a 33 z)a 3 . 

It is evident then that if we transform the e's by a non- 
singular linear homogeneous transformation, the coeffi- 
cients of the new basis hypernumbers, a, are the transforms 
of the original coefficients under the contragredient trans- 
formation. 

Inasmuch as the transformation is linear, the transform of 
a sum will be the sum of the transforms of the terms of the 
original sum. The transformation as a geometrical process 
is equivalent to changing the axes. This process evidently 
gives us a new triple, but must be considered not to give 
us a new hypernumber nor a new vector. Indeed, a vector 
cannot be defined by a triple of numbers alone. There 
is also either explicitly stated or else implicitly understood 
to be a basis, or on the geometric side a definite set of axes 
such that the triple gives the components of the vector 
along these axes. It is evident that the success of any 
system of vector calculus must then depend upon the 
choice of modes of combination which are not affected by 
the change from one basis to another. This is the case 
with addition as we have defined it. We assume that we 
may express any vector or hypernumber in terms of any 
basis we like, and usually the basis will not appear. 

If the transformation is such as to leave the angles be- 



ADDITION OF VECTORS 55 

tween ei, e 2 , e 3 the same as those between a\, a 2 , a 3 , the 
second trihedral being substantially the same as the first 
rotated into a new position, with the lengths in each case 
remaining units, then the transformation is called orthog- 
onal. We may define an orthogonal transformation algebra- 
ically as one such that if followed by the contragredient 
transformation the original basis is restored. 

4. Differential of a Vector. If we consider two points 
at a small distance apart, the vector to one being p, to the 
other p', and the vector from the first to the second, Ap 
= p' — p, where Ap = As-e, e being a unit vector in the 
direction of the difference, we may then let one point ap- 
proach the other so that in the limit e takes a definite posi- 
tion, say a, and we may write ds for As, and call the result 
the differential of p for the given range over which the p f 
runs. In the hypernumbers we likewise arrive at a hyper- 
number 

dp = dxei -f- dye?, + dzez, 

where now ds is a linear homogeneous irrational function 
of dx, dy, dz, which = V (dx 2 + dy 2 + dz 2 ) in case e ly e 2 , e 3 
form a trirectangular system of units. 

The quotient dpjdt is the velocity at the point if t repre- 
sents the time. The unit vector a: is the unit tangent for 
a curve. We generally represent the principal normal and 
the binormal by jS, 7 respectively. When p is given as 
dependent on a single variable parameter, as t for instance, 
then the ends of p may describe a curve. We may have 
in the algebraic form the coordinates of p alone dependent 
upon the parameter, or we may have both the coordinates 
and the basis dependent upon t. For instance, we may ex- 
press p in terms of ei, e 2 , e 3 which are not dependent upon 
t but represent fixed directions geometrically, or we may 
express p in terms of three hypernumbers as w, r, J* which 



56 VECTOR CALCULUS 

themselves vary with t, such as the moving axes of a system 
in space. In relativity theories the latter method of repre- 
sentation plays an important part. 

5. Integral of a Vector. If we add together n vectors and 
divide the result by n we have the mean of the n vectors, 
which may be denoted by p. If we select an infinite 
number of vectors and find the limit of their sum after 
multiplication by dt, the differential of the parameter by 
which they are expressed, such limit is called the integral 
of the vector expressed in terms of t, and if we give t two 
definite values in the integral and subtract one result from 
the other, the difference is the integral of the vector from 
the first value of t to the second. More generally, if we 
multiply a series of vectors, infinite in number, by a corre- 
sponding series of differentials, and find the limit of the 
sum of the results, such limit, when it exists, is called the 
integral of the series. In integration, as in differentiation, 
the usual difficulties met in analysis may appear, but as 
they are properly difficulties due to the numerical system 
and not to the hypernumbers, we will suppose that the 
reader is familiar with the methods of handling them. 

The mean in the case of a vector which has an infinite 

sequence of values is the quotient of the integral taken on 

some set of differentials, divided by the integral of the set 

of differentials itself. The examples will illustrate the 

use of the mean. 

EXAMPLES 

(1) The centroid of an arc, an area, or a volume is found 
by integrating the vector p itself multiplied by the dif- 
ferential of the arc, ds, or of the surface, du, or of the volume 
dv. The integral is then divided by the length of the 
arc, the area of the surface, or the volume. That is 

- Sheets ffpdu • fffpdv m 

P — — , or - — or — 

b— a A V 



ADDITION OF VECTORS 



57 



(2) An example of average velocity \s found in the following 
(Bjerknes, Dynamic Meteorology, Part II, page 14) obser- 
vations of a small balloon. 



2 = Ht. in 

Meters 


Az 


Direction 


Velocity 

(w/sec.) 


Products 


77 
680 
960 
1240 
1530 
1810 
2090 
2430 
2730 
3040 
3400 
3710 
4030 
4400 


603 
280 
280 
290 
280 
280 
340 
300 
310 
360 
310 
320 
370 


S. 50° E. 
S. 57° E. 
S. 36° E. 
S. 28° W. 
S. 2°W. 
S. 2°W. 
S. 35° W. 
S. 53° W. 
S. 69° W. 
S. 55° W. 
S. 53° W. 
S. 58° W. 
S. 37° W. 


3.4 
4.0 
5.3 
1.5 
1.8 
2.0 
1.5 
1.8 
1.8 
3.0 
2.8 
4.4 
10.2 


2050 

1120 

1484 

435 

504 

560 

510 

540 

558 

1080 

868 

1408 

3773 



To average the velocities we notice that on the assump- 
tion that the upward velocity was uniform the distances 
vertically can be used to measure the time. We therefore 
multiply each velocity by the difference of elevations 
corresponding, the products being set in the last column. 
These numbers are then taken as the lengths of the vectors 
whose directions are given by the third column. The 
sum of these is found graphically, and divided by the total 
difference of distance upward, that is, 4323. In the same 
manner we can find graphically the averages for each 1000 
meters of ascent. We may now make a new table in order 
to find other important data, as follows : 



Height 


. Pressure 
(ra-bars) 


Dens, (ton/w 3 ) 


Veloc. . 


Spec. Mo- 
mentum 
(ton/ra 2 sec.) 


4000 

3000 

2000 

1000 

75 


622 
705 
797 
899 
1003 


0.00083 
0.00092 
0.00102 
0.00112 


3.8 
1.6 

• 2.4 
3.7 


0.0032 
0.0015 
0.0025 
0.0041 



;,s 



VECTOR CALCULUS 



We now find the average velocity between the 1000 m-bar, 
the 900 ra-bar, the 800 m-bar, the 700 ra-bar, and the 600 
ra-bar. The direction is commonly indicated by the in- 
tegers from to 63 inclusive, the entire circle being divided 
into 64 parts, each of 5f°. East is 0, North is 16, NW. is 
24, etc. The following table is found. 



Pressure 


Height 


Spec. Vol. 

(m 3 /Ton) 


Direction 


Veloc. 


Spec. Mo- 
mentum 


600 

700 

800 

' 900 

1000 

1002.6 


4274 

3057 

1970 

989 

99 

76 


1217 

1087 

981 

890 

890 


8 

7 

20 

25 

25 


5.2 
1.7 
2.4 
3.7 
3.4 


0.0043 
0.0016 
0.0024 
0.0042 
0.0040 



Of course, specific momenta should be averaged like veloc- 
ities but usually owing to the rough measurements it is 
sufficient to find specific momenta from the average 
velocities. 



ADDITION OF VECTORS 



59 



EXERCISES 

1. Average as above the following observations taken at places 
mentioned (Bjerknes, p. 20), July 25, 1907, at 7 a.m. Greenwich time. 



Isobar 


Dyn. Ht. 


Az 


Direction 


Veloc. 




100 

200 

300 

400 

500 

600 

700 

800 

900 
1000 
1001.2 


16374 

11947 

9320 

7301 

5648 

4240 

3020 

1938 

975 

107 

98 


4427 

2627 

2019 

1653 

1408 

1220 

1082 

963 

867 

9 




10 

18 

19 

8 

5 

4 

4 

36 

35 


4.7 
3.2 
3.4 
3.3 
2.6 
2.5 
2.5 
1.4 
4.5 
4.5 


Uccle, 
Lat. 50° 48' 
Long. 4° 22' 


100 
200 
300 
400 
500 
600 
700 
800 
900 
955.9 


16238 

11817 

9240 

7248 

5626 

4244 

3038 

1955 

977 

471 


4421 
2577 
1992 
1622 
1382 
1206 
1083 
978 
506 


3 
6 
7 
2 
3 
2 

62 
4 

30 


10.0 
6.5 
7.6 

10.2 
6.7 
6.8 
5.3 
0.6 
2.1 


Zurich, 
Lat. 47° 23' 
Long. 8° 33' 


200 
300 
400 
500 
600 
700 
800 
900 
1000 


11890 

9241 

7240 

5643 

4196 

2991 

1927 

981 

118 

17 


2649 

2001 

1597 

1447 

1205 

1064 

946 

863 

101 


59 
57 
58 
55 
49 
41 
38 
56 
55 


9.2 
10.5 
8.8 
8.0 
2.9 
2.9 
1.9 
4.3 
3.4 


Hamburg, 
Lat. 53° 33' 
Long. 9° 59' 



GO 



VECTOR CALCULUS 



2. If the direction of the wind is registered every hour how is the 
average direction found? Find the average for the following observa- 
tions. 



Station 


Pikes 
Peak 


Vienna 


Mauritius 


Cordoba 


S Orkneys 


Elev 


4308 m. 


26 m. 


15 m. 


437 m. 


25 m 








Summer 


Winter 


Dec-Feb. 


Winter 


Summer 


Time 












Vel. 


Az. 


Vel. 


Az. 


Vel. 


Az. 


Vel. 


Az. 


Vel. 


Az. 


a.m 


0.84 
1.34 
1.46 
1.05 
0.43 
0.66 
1.03 


100° 
83 
71 
57 
12 
279 
262 


0.47 
0.56 
0.42 
0.33 
0.22 
0.17 
0.36 


62 
61 
59 
57 
46 
303 
257 


1.00 
1.30 
1.30 
1.00 
1.10 
1.80 
2.40 


100 
97 
98 
119 
241 
312 
326 


0.94 
1.06 
1.44 
2.03 
2.17 
0.50 
2.78 


115 
111 
121 
132 
136 
252 
314 


0.52 
0.52 
0.51 
0.51 
0.52 
0.53 
0.54 


70 


2 


51 


4 


30 


6 


5 


8 


343 


10 


?m 


12 noon 


255 


14 


1.09 


256 


0.58 


242 


1.60 


332 


3.56 


315 


0.54 


245 


16 


0.95 


253 


0.64 


232 


1.30 


304 


3.36 


305 


0.54 


42 


18 


0.74 


247 


0.47 


223 


0.20 


10 


1.75 


299 


0.53 


35 


20 


0.49 


47 


0.14 


186 


0.90 


101 


0.72 


44 


0.53 


50 


22 


0.36 


153 


0.25 


72 


1.00 


102 


0.89 


128 


0.52 


60 



Bigelow, Atmospheric Circulation, etc., pp. 313-315. 

3. The following table gives the mean magnetic deflecting vectors, 
in four zones, the intensity measured in 10~ 6 dynes, <p measured from 
S. to E., N., W., and is measured above the horizon. The vector is 
the deflection from the mean position. Find the average for each 
zone. (Bigelow, pp. 324-325.) 



Time 


Arctic 


N Temperate 


Tropic 


S Temperate 




























s 


e 


<P 


s 





<P 


s 





<p 


s 





<p 


a.m. 





























60 


-36° 


345° 


15 


-30° 


111° 


20 


-33° 


5° 


19 


27° 


259° 


1 


63 


-44 


355 


14 


-35 


109 


19 


-32 


16 


19 


31 


250 


2 


69 


-43 


5 


14 


-32 


102 


20 


-36 


7 


17 


35 


251 


3 


74 


-44 


16 


14 


-33 


108 


20 


-42 


6 


18 


36 


243 


4 


75 


-42 


25 


15 


-35 


112 


18 


-34 


10 


20 


36 


226 


5 


77 


-42 


30 


17 


-33 


110 


17 


-37 


6 


21 


33 


223 


6 


78 


-40 


32 


20 


-31 


112 


19 


-36 


4 


24 


31 


222 


7 


76 


-40 


36 


22 


- 6 


107 


21 


-37 


339 


26 


24 


235 


8 


65 


-37 


45 


25 


3 


99 


24 


-30 


297 


28 


28 


248 


9 


54 


-18 


68 


26 


24 


66 


26 


23 


228 


28 


33 


256 


10 


39 


31 


117 


27 


37 


49 


35 


25 


210 


26 


-27 


296 


11 


47 


44 


195 


25 


38 


312 


43 


22 


204 


25 


-37 


327 



ADDITION OF VECTORS 61 

4. Find the resultant attraction at a point due to a segment of a 
straight line which is (a) of uniform density, (6) of density which varies 
as the square of the distance from one end. What is the mean attrac- 
tion in each case? 

5. Show that p = ta + \P$ is the equation of a parabola, that the 
equation of the tangent is p = Ua + \t\ 2 & + x(a + ttfi), that tangents 
from a given point are given by t = p ± V (p 2 — 2q), the point being 
pa + q/3, the chord of contact is p = — qP + y(a + PP) which has a 
direction independent of q so that all points of the line p = pa + zP 
have corresponding chords of contact which are parallel. If a chord 
is to pass through the point aa + bp for differing values of p, then 
q = ap — b and the moving point pa + qP lies on the line p — pa 
-f- (ap — b)P, whose direction is independent of b. 

6. If a, /S, 7 are vectors to three collinear points, then we can find 
three numbers a, b, c such that 

aa + 6/S + cy = = a + b + c. 

7. In problem 5 show that if three points are taken on the parabola 
corresponding to the values t\, U, tz, then the three points of intersection 
of the sides of the triangle they determine with the tangents at the 
vertices of the triangle are collinear. 

8. Determine the points that divide the segment joining A and B, 
points with vectors a and 0, in the ratio I : m, both internally and ex- 
ternally. Apply the result to find the polar of a point with respect to 
a given triangle, that is, the line which passes through the three points 
that are harmonic on' the three sides respectively with the intersection 
of a line through the given point and the vertex opposite the side. 

9. Show how to find the resultant field due to superimposed fields. 

10. A curve on a surface is given by p = u(u, v), u = /(v), study the 
differential of p. 



CHAPTER V 

VECTORS IN A PLANE 
1. Ratio of Two Vectors. We purpose in this chapter to 
make a more detailed study of vectors in a plane and the 
hypernumbers corresponding. In the plane it is convenient 
to take some assigned unit vector as a reference for all 
others in the plane, though this is not at all necessary in 
most problems. In fact we go back for a moment to the 
fundamental idea underlying the metric notion of number. 
According to this a number is defined to be the ratio be- 
tween two quantities of the same concrete kind, such as 
the ratio of a rod to a foot. If now we consider the ratio 
of vectors, regarding them as the same kind of quantity, 
it is clear that the ratio will involve more than merely 
numerical ratio of lengths. The ratio in this case is in 
fact what we have called a hypernumber. For every pair 
of vectors p, x there exists a ratio p : x and a reciprocal 
ratio x : p. This ratio we will designate by a roman 
character 

P 



p : x = p/x = 



IT 



That is to say, we may substitute p for qw. 

2. Complex Numbers. If we draw p and x from one 
point, they will form a figure which has two segments for 
sides and an angle. (In case they coincide we still con- 
sider they have an angle, namely zero.) In this figure p is 
the initial side and x is the terminal side. Then their 
complex ratio is x : p. Since this ratio is to be looked upon 
as a multiplier, it is clear that if we were to reduce the 
sides in the same proportion, the ratio would not be changed. 

62 



VECTORS IN A PLANE 63 

A change of angle would, however, give a different ratio. 
However, we will agree that all ratios are to be considered 
as equivalent, or as we shall usually say, equal, not only 
when the figures to which they correspond have sides in 
the same proportion, but also when they have the same 
angles and sides in proportion, even if not placed in the 
plane in the same position. For instance, if the vectors 
AB, AC make a triangle which is similar to the triangle 
DE, DF, if we take the sides in this order, then we shall 
consider that whatever complex or hypernumber multiplies 
AC into AB will also multiply DF into DE. This axiom 
of equivalence is not only important but it differentiates 
this particular hypernumber from others which might just 
as well be taken as fundamental. For instance, the Gibbs 
dyad of t : p is equally a hypernumber, but we cannot 
substitute for ir or p any other vectors than mere multiples 
of 7r or p. It is clear that in the Gibbs dyad we have a 
more restricted hypernumber than in the ordinary com- 
plex number which has been just defined, and which is a 
special case of the Hamiltonian quaternion. If we have 
a Gibbs dyad q, we can find the two vectors ir and p save 
as to their actual lengths. But with the complex number 
q we cannot find ir and p further than to say that for every 
vector there is another in the ratio q. In other words the 
only transformations allowed in the Gibbs dyad are transla- 
tion of the figure AB, AC or magnification of it. In the 
Hamiltonian quaternion, or complex number, the trans- 
formations of the figure AB, AC may be not only those 
just mentioned but rotation in the plane. 

In order to find a satisfactory form for the hypernumber 
q which we have characterized, we further notice that if 
we change the length of x in the ratio m then we must 
change q in the same ratio, and if we set for the ratio of the 



64 VECTOR CALCULUS 

length or intensity of w to that of p the number r, it is evi- 
dent that we ought to take for q an expression of the form 

q = r<p(0), 

where <p(6) is a function of 0, the angle between p and t, 
only. Further if we notice that we now have 

7T = r(p(6)p, 

where the first factor affects the change of length, the 
second the change of direction, it is plain that for a second 
multiplication by another complex number q' = r'<p(0'), 
we should have 

tt' = r , rcp(e , )i P {e) P = r'rip(W + 6)p. 

Whence we must consider that 

viO'Md) = <p{e f +$) = view). 

These expressions are functions of two ordinary numerical 
parameters, 0, 0', and are subject to partial differentiation, 
just like any other expressions. Differentiating first as to 
0, then as to 6', we find (<p f being the derivative) 

<p\eM6') = ?'($+ e f ) = wmb), 

whence 

. v y) = V '{6') _ 

where & is a constant and does not depend upon the angle at 
all. It may, however, depend upon the plane in which 
the vectors lie, so that for different planes A; may be, and 
in fact is, different. N 

Since, when = the hypernumber becomes a mere 
numerical multiplier, 

<p'(0) = MO). 

If now we examine the particular function 
<p(0) = cos 0+k sin 6, 



VECTORS IN A PLANE 65 

which gives 

<p'(d) = — * sin $ + k cos 6 = k cos 6 + k 2 sin 6, 

we find all conditions are satisfied if we take k 2 = — 1. 
We may then properly use this function to define <p. 
This very simple condition then enables us to define hyper- 
numbers of this kind, so that we write 

q = r(cos 6 + k sin 9) = r cks 6 = r g , 

where k 2 = — 1. 

3. Imaginaries. It is desirable to notice carefully here 
that we must take k 2 equal to — 1, the same negative 
number that we have always been using. This is important 
because there are other points of view from which the 
character of k and k 2 would be differently regarded. For 
instance, in the original paper of Hamilton, On Algebraic 
Couples, the k, or its equivalent, is regarded as a linear 
substitution or operator, which converts the couple (a, b) 
into the couple (— b, a). While it is true that we may so 
regard the imaginary, it becomes at once obvious that we 
must then draw distinctions between 1 as an operator, and 
1 as a number, and so for — 1, and indeed for any expression 
x + yi. In fact, such distinctions are drawn, and we find 
these operators occasionally called matrix unity, etc. From 
the point of view of the hypernumber, this distinction is 
not possible. Hypernumbers are extensions of the number 
system, similar to radicals and other algebraic numbers. 
The fact that, as we will see later, they are not in general 
commutative, does not prevent their being an extension. 

4. Real, Imaginary, Tensor, Versor. In the complex 
number 

q = r cos 6 + r sin 6 • k 

the term r cos 6 is called the real part of q and may be written 
Rq. The term r sin 6-k is called the imaginary part of q 



66 VECTOR CALCULUS 

and written Iq. The number r is called the tensor of q and 
written Tq. The expression cos 6 + sin • k is called the 
versor of </ and written Uq. Therefore, 

q= Rq+ Iq= TqUq. 

If q appears in the form q — a + bk we see at once that 

Rq = a , Iq= bk, Tq = V (a 2 + b 2 ), 6 = taiT^/a. 

5. Division. If we have w = qp, then we also write 
p = g -1 7r. It becomes evident that 

&T l = RqKTqf, Iq-' = - Iql(Tq)\ Tq-* = 1/Tq, 
Uq- 1 = cos 6 — sin 6 • k. 

6. Conjugate, Norm. The hypernumber q = Kq — Rq 
— Iq is called the conjugate of q. If q belongs to the figure 
AB, AC, then q belongs to an inversely similar triangle, that 
is, a similar triangle which has been reflected in some 
straight line of the plane. The product q° = Nq = (Tq) 2 
is called the norm of q. It also has the name modulus of q, 
particularly in the theory of functions of complex variables. 

Evidently, 

Rq = i(q + q), Iq = h(q - ~q), r 1 = W* ^q~ l = Uq- 

7. Products of Complex Numbers. From the definitions 
it is clear that the product of two complex numbers q, r, 
is a complex number s, such that 

Ts = TqTr,_ ZJ= zq+ Zr, 
Rqr = Rrq = Rqr = Rrq = RqRr - Tlqlr, 
Rqr = Rqr = Rrq = Rrq = RqRr + Tlqlr, 
Iqr = Irq = — Tqr = — Irq = Rqlr + Rrlq, 
Iqr = Irq = — Iqr = — Irq = Rrlq — Rqlr. 

Hence if Rqr = 0, the angles of q and r are complementary 
or have 270° for their sum. 



VECTORS IN A PLANE 67 

If Rqr = 0, the angles differ by 90°. In particular 

we may take r = 1. 
If Iqr = 0, the angles are supplementary or opposite. 
If Iqr = 0, the angles are equal or differ by 180°. 

8. Continued Products. We need only to notice that 



(qrs- • -z) = (z- • -srq). 

It is not really necessary to reverse the order here as the 
products are commutative, but in quaternions, of which 
these numbers are particular cases, the products are not 
usually commutative, and the order must be as here 
written. 

9. Triangles. If ft y, 5, e are vectors in the plane, and 

e = gft 5 = gy f 

then the triangle of ft e is similar to that of y, 5, while if 
e = gft 5 = ?7, 

the triangles are inversely similar. 

These equations enable us to apply complex numbers to 
certain classes of problems with great success. 

10. Use of Complex Numbers as Vectors. If a vector a 
is taken as unit, every vector in the plane may be written 
in the form qa, for some properly chosen q. We may 
therefore dispense with the writing of the a, and talk of 
the vector q, always with the implied reference to a certain 
unit a. This is the well-known method of Wessel, Argand, 
Gauss, and others. However, it should be noticed that 
we have no occasion to talk of q as a point in the plane. 

EXAMPLES 
(1) Calculate the path of the steam in a two-wheel tur- 
bine from the following data. The two wheels are rigidly 
connected and rotate with a speed a = 400 ° ft./sec. Be- 



68 VECTOR CALCULUS 

tween them are stationary buckets which turn the exhaust 
steam from the buckets of the first wheel into those of the 
second wheel. The friction in each bucket reduces the speed 
by 12%. The steam issues from the expansion nozzle at a 
speed of /3 = 2200 2 o°. The proper exhaust angles of the 
buckets are 24°, 30°, 45°. Find the proper entrance angles 
of the buckets. 

7 = relative velocity of steam at entrance to first wheel. 
= 2200 20 - 400o = 1830 24 .3. 

8 = velocity of issuing steam, 88% of preceding, 

= 1610x56. 

€ = entrance velocity to stationary bucket. 

= 5 + a = I6IO1M + 400o = 1255i4 8 .4. 
f = exit - 1105 30 . 
= entrance to next bucket = £ — a = 1105 30 — 400o 

= 78044.3. 
77 = exit = 69O135. Absolute exit velocity = 690i35 
+ 400 = 495ioo. 

Steinmetz, Engineering Mathematics. 

(2). We may suppose the student is somewhat familiar 
with the usual elementary theory of the functions of a 
complex variable. If w is an analytic function of z, both 
complex numbers, then the real part of w, Rw, considered 
as a function of x, y or u, v, the two parameters which de- 
termine z, will give a system of curves in the x, y, or the 
u, v plane. These may be considered to be the transforma- 
tions of the curves Rw = const, which are straight lines 
parallel to the Y axis in the w plane. Similarly for the 
imaginary part. The two sets will be orthogonal to each 

other, since the slope of the first set will be ^— / -z — ; 

J * 1 1 dTIw/dTIw _ 

and 01 the other set ^ — / —^ — . But these are 

ox I dy 



VECTORS IN A PLANE 69 

negative reciprocal, since 

dRw dTIw dRw_ dTIw 

~ — n and ~ — ~ 

ox oy ay ox 

EXERCISES 

1 . If a particle is moving with the velocity 12028° and enters a medium 
which has a velocity given by 

<r = P + 36 sin z [p, 0] 8 °, 
what will be its path? 

2. The wind is blowing steadily from the northwest at a rate of 
16 ft. /sec. A boat is carried round in circles with a velocity 12 ft. /sec. 
divided by the distance from the center. The two velocities are com- 
pounded, find the motion of the boat if it starts at the point p = 4 °. 

3. A slow stream flows in at the point 12 ° and out at the point 
12i8o°, the lines of flow being circles and the speed constant. A chip 
is floating on the stream and is blown by the wind with a velocity 
640 . Find its path. 

4. If a triangle is made with the sides q, r then R.qr is the power of 
the vertex with reference to the circle whose diameter is the opposite 
side. The area of the triangle is \TIqr. 

5. The sum q + r can be found by drawing vectors qa, ra. 

6. How is qr constructed? qr? 

7. If OAE is a straight line and OCF another, and if EC and AF 
intersect in B, then OA BC + OC • AB + OB • CA = 0. If 0, A, B, C 
are concyclic this gives Ptolemy's theorem. 

8. If ABC is a triangle and LM a segment, and if we construct 
LMP similar to ABC, LMQ similar to BCA, and LMR similar to CAB, 
then PQR is similar to CAB. 

9. If the variable complex number u depends on the real number x 
as a variable parameter, by the linear fractional form 

ax + b 

u - 



ex + d 

then for different values of x the vector representing u will terminate on 
a circle. 

For if we construct 

b 

U ~d ' 

w = 

a 

u 

c 



70 VECTOR CALCULUS 

this reduces to — (cx/d), hence the angle of w, which is the angle between 
u — ale and u — b/d, is the angle of — d/c and is therefore constant. 
Hence the circle goes through a/c (x = «) and b/d (x = 0). 

10. If 

_ x(c — b)a + b(a — c) , 
U ~ k(c-b) + (a- c) 

where x is a variable real parameter, then the vector representative of 
u will terminate on the circle through A, B, C, where OA represents 
a, OB represents 6, and OC represents c. 

11. Given three circles with centers C 1} d, C3, and O their radical 
center, P any point in the plane, then the differences of the powers of 
P with respect to the three pairs of circles are proportional respectively 
to the projections of the sides of the triangle CiC 2 Cz on OP. 

12. Construct a polygon of n sides when there is given a set of points, 
Ci, C2, - • -, C n which divide the sides in given ratios a x : bi, a 2 : 62, • • •, 
a» : 6„. 

If the vertices are &, £ 2 , • • • , in, and the points Ci, C 2 , • • • , C n are 
at the ends of vectors 71, 72, •••, y n , we have 

Olll + &lfc = 7l(ai +6l) ' * * CLntn + bnh = 7n(a n + b n ). 

The solution of these equations will locate the vertices. When is the 
solution ambiguous or impossible? 

13. Construct two directly similar triangles whose bases are given 
vectors in the plane, fixed in position, so that the two triangles have a 
common vertex. 

14. Construct the common vertex of two inversely similar triangles 
whose bases are given. 

15. Construct a triangle ABC when the lengths of the sides AB and 
AC are given and the length of the bisector AD. 

1G. Construct a triangle XYZ directly similar to a given triangle 
PQR whose vertices shall be at given distances from a fixed point 0. 

Let the length of OX be a, of OY be 6, and of OZ be c. Then X is 
anywhere on the circle of radius a and center O. We have XY/XZ 
= PQIPR, that is, 

OY -OX = PQ 
OZ-OX PR' 
whence we have 

OXQR + OYRP + OZPQ = 0. 
We draw OXK directly similar to RPQ giving KO/OX = QR/RP and 

KO + OY + OZ -£§ = 0, that is, 



VECTORS IN A PLANE 71 

In KOY we have the base KO and the length OY = b, and length of 

_ length PQ 
length RP' 

We can therefore construct KOY and the problem is solved. 

17. The hydrographic problem. Find a point X from which the three 
sides of a given triangle ABC are seen under given angles. 

XB/XA = y cks 0, XC/XA = z cks p. 

XB = XA + AB, XC = XA + AC. 
Eliminate XA giving 2 cks <?•# A + y cksd-AC = BC. Find U such 
that z AJBI7 = Z AXC, Z ACtf = Z AXB, then BU = z cks *>. 
BA,CU = y cks OCA. 

Construct A ACX directly similar to A A UB. 

18. Find the condition that the three lines perpendicular to the 
three vectors pa, qa, ra at their extremities be concurrent. 

We have p + xkp = q + ykq = r + zkr. Taking conjugates 
q — xkp = p — ykq = r— zkr. Eliminate x, y, z from the four 
equations. 

19. If a ray at angle is reflected in a mirror at angle a the reflected 
ray is in the direction whose angle is 2 a — /3. Study a chain of mirrors. 
Show that the final direction is independent of some of the angles. 

20. Show that if the normal to a line is a and a point P is distant y 
from the line, and from P as a source of light a ray is reflected from the 
line, its initial direction being — qa, then the reflected ray has for 
equation — 2ya + tqa = p. 

For further study along these lines, see Laisant: Theorie et 
Application des Equipollences. 

11. Alternating Currents. We will notice an application 
of these hypernumbers to the theory of alternating currents 
and electromotive forces, due to C. P. Steinmetz. 

If an alternating current is given by the equation 

I = Io cos 2wf(t - h), 

the graph of the current in terms of t is a circle whose 
diameter is 7 making an angle with the position for t = 
of 2wfti. The angle is called the phase angle of the current. 
If two such currents of the same frequency are superim- 



72 VECTOR CALCULUS 

posed on the same circuit, say 



we may set 



7 = 7 cos 2irf(t - ti), 
F = Jo' cos 2tt/(* - fcO, 
sex 

7 cos 2vfh + h' cos 2tt/V = 7 " cos 2wfh, 

7 sin 2tt/<i + U sin 2u//i' = 7 " sin 2irft 2t 

7" = 7 " cos 2wf(t - it), 

which also has for its graph a circle, whose diameter is the 
vector sum of the diameters of the other two circles. We 
may then fairly represent alternating currents of the simple 
type and of the same frequency by the vectors which are 
the diameters of the corresponding circles. The same 
may be said of the electromotive forces. 

If we represent the current and the electromotive force 
on the same diagram, the current would be indicated by a 
yellow vector (let us say) traveling around the origin, 
with its extremity on its circle, while at the same time the 
electromotive force would be represented by a blue vector 
traveling with the same angular speed around a circle 
with a diameter of different length perhaps. The yellow 
and the blue vectors would generally not coincide, but they 
would maintain an invariable angle, hence, if each is con- 
sidered to be represented by a vector, the ratio of these 
vectors would be such that its angle would be the same for 
all times. This angle is called the angle of lag, or lead, 
according as the E.M.F. is behind the current or ahead of it. 

The law connecting the vectors is 

E= ZI, 

where E is the electromotive force vector, that is, the vector 
diameter of its circle, 7 is the current vector, the diameter 
of its circle, and Z is a hypernumber called the impedance, 



VECTORS IN A PLANE 73 

[<p/0], measured in ohms. The scalar part of Z is the 
resistance of the circuit, while the imaginary part is the 
reactance, the formula for Z being 

Z = r — xk. 

The value of x is 2irfL, where/ is the frequency, [T~ l ], and 
L is the inductance, [^G -1 ? 1 ], in henry s, or — l/2irfC where 
C is the permittance, [OT 1 ^ -1 ], in farads. [1 farad = 9- 10 11 
e.s. units = 10 -9 e.m. units, and 1 ^nn/ = ^lO -11 e.s. 
units = 10 9 e.m. units.] It is to be noticed that reactance 
due to the capacity of the circuit is opposite in sign to 
that due to inductance. 

The law above is called the generalized Ohm's law. We 
may also generalize KirchofFs laws, the two generalizations 
being due to Steinmetz, and having the highest importance, 
inasmuch as by the use of these hypernumbers the same 
type of calculation may be used on alternating circuits as 
on direct circuits. The generalization of KirchofFs laws 
is as follows : 

(1) The vector sum of all electromotive forces acting in a 
closed circuit is zero, if resistance and reactance electro- 
motive forces are counted as counter electromotive forces. 

(2) The vector sum of all currents directed toward a 
distributing point is zero. 

(3) In a number of impedances in series the joint im- 
pedance is the vector sum of all the impedances, but in a 
parallel connected circuit the joint admittance (reciprocal 
of impedance) is the sum of the several admittances. 

The impedance gives the angle of lag or lead, as the angle 
of a hyper number of this type. 

We desire to emphasize the fact that in impedances we 
have physical cases of complex numbers. They involve 
complex numbers just as much as velocities involve positive 



74 VECTOR CALCULUS 

of negative velocity, or rotations involve positive or nega- 
tive. We may also affirm that the complex currents and 
electromotive forces are real physical existences, every 
current implying a power current and a wattless current 
whose values lag 90° (as time) behind the power current. 
The power electromotive force is merely the real part of 
the complex electromotive force, and the wattless E.M.F. the 
imaginary part of the complex electromotive force, both 
being given by the complex current and the complex 
impedance. 

We find at the different points of a transmission line that 
the complex current and complex electromotive force satisfy 
the differential equations 

dl/ds = (g + Cok)E, dE/ds = (r + Look)L 

The letters stand for quantities as follows: g is mhos I mile, 
r is ohms/mile, C is farads/mile, L is henrys/mile. co = 2irf. 
Setting 

m* = (r + Lo>k)(g + Cirk), I 2 = (r + Lak)/(g + Cwk), 
so that m is [X -1 ] while / is ohms/mile, the solution of the 
equations is 

E = E cosh ms + ll sinh ms, 
I = Iq cosh ms + 1~ 1 Eq sinh ms, 

where E and 7 are the initial values, that is, where s = 0. 
If we set Eq = ZqIq and then set Z = Z cosh h, I = 
Z sinh h we have 

E = Z cosh (ms + h)I , I = l~ l Z sinh (ms + h)I , 

E = I coth (ms + h)I, 
E = sech h cosh (ms + h)E , 
I = csch h sinh (ms + h)I . 

To find where the wattless current of the initial station has 
become the power current we set I = kl , that is, 

sinh (ms -f- h) = k sinh h. 



VECTORS IN A PLANE 75 

The value of s must be real. 

EXAMPLES 
(1) Let r = 2 ohms/mile, L = 0.02 henrys/mile, 

C = 0.0000005 farads/mile, 
g = 0, to = 2000, coL = 40 ohms/mile, conductor 

reactance, 
r + Look = 2 + 40/c ohms/mile impedance 

= 40.5 87 .i5 o . 
uC = 0.001 mhos/mile dielectric susceptance. 
g + Coik = 0.001 k mhos/mile dielectric admit- 
tance = 0.001 90 °. 
(g + Cuk)~ l = 1000/j" 1 = 1000 27 o° ohms/mile 

dielectric impedance. 
m 2 = 0.0405i 77 .i5°, m = 0.2001 88 .58°, 
P = 40500_.2.85°, I = 201.25_i.43°. 

Let the values at the receiver (s = 0) be 

E = 1000 o volts, 7 = o. 
Then we have E = 1000 cosh s0.2001 8 8.58°, 

for s = 100 E = 1000 cosh 20.01 88 . 58 = 625.9 45 .92°, 

I = 2.77 27 o, 
for s = 8 E = 50.01i26.ot, 
for s = 16 E = 1001i 80 .3°, 
for s = 15.7 E = 1000i 8 o°, a reversal of phase, 
for s = 7.85 E = 90 o. 

At points distant 31.4 miles the values are the same. 
If we assume that at the receiver end a current is to be 
maintained with 

Jo = 50 40 ° with E = 1000 o, 

E = 1000 cosh s0.2001 88 . 58 ° + 10062 38 . 5 7° sinh s0.2001 88 . 5 8°, 
I = 50 4 o° cosh sm + 5i. 4 3° sinh sm. 
At s = 100 E = 10730n355°. 

MacMahon, Hyperbolic Functions. 



76 VECTOR CALCULUS 

(2) Let E - 10000, 7 - 65i 3 . 5 ° r = 1, g = 0.00002 

Ceo = 0.00020 period 221.5 miles, o>L = 4. 

(3) The product P = EI represents the power of the 
alternating current, with the understanding that the fre- 
quency is doubled. The real or scalar part is the effective 
power, the imaginary part the wattless or reactive power. 
The value of TP is the total apparent power. The cos z P 
is the power factor, and sin / P is the induction factor. 
The torque, which is the product of the magnetic flux by 
the armature magnetomotive force times the sine of their 
angle is proportional to TIP, where E is the generated 
electromotive force, and/ is the secondary current. In 
fact, the torque is TI'EI-p/2irf where p is the number of 
poles (pairs) of the motor. 

12. Divergence and Curl. In a general vector field the 
lines have relations to one another, besides having the 
peculiarities of the singularities of the field. The most 
important of these relations depend upon the way the lines 
approach one another, and the shape and position of a 
moving cross-section of a vector tube. There is also at 
each point of the field an intensity of the field as well as a 
direction, and this will change from point to point. 

Divergence of Plane Lines. If we examine the drawing 
of the field of a vector distribution in a plane, we may 
easily measure the rate of approach of neighboring lines. 
Starting from two points, one on each line, at the intersec- 
tion of the normal at a point of the first line and the second 
line, we follow the two lines measuring the distance apart 
on a normal from the first. The rate of increase of this 
normal distance divided by the normal distance and the 
distance traveled from the initial point is the divergence of 
the lines, or as we shall say briefly the geometric divergence 
of the field. It is easily seen that in this case of a plane 




VECTORS IN A PLANE 77 

field it is merely the curvature of the curves orthogonal 
to the curves of the field. 

For instance, in the figure, the tangent to a curve of the 
field is a, the normal at the same point /5. The neighboring 
curve goes through C. The differential of the normal, 
which is the difference of BD and 
AC, divided by AC, or BD, is the 
rate of divergence of the second curve 
from the first for the distance AB. 
Hence, if we also divide by AB we 
will have the rate of angular turn of 
the tangent a in moving to the neigh- 
boring curve, the one from C. This rate of angular turn 
of the tangent of the field is the same as the rate of turn of 
the normal of the orthogonal system, and is thus the curva- 
ture of the normal system. 

Curl of Plane Lines. If we find the curvature of the 
original lines of the field we have a quantity of much im- 
portance, which may be called the geometric curl. This 
must be taken plus when the normal to the field on the 
convex side of the curve makes a positive right angle with 
the tangent, and negative when it makes a negative right 
angle with the tangent. Curl is really a vector, but for 
the case of a plane field the direction would be perpendicular 
to the plane for the curl at every point, and we may con- 
sider only its intensity. 

Divergence of Field. Since the field has an intensity as 
well as a direction, let the vector characterizing the field 
be cr = Ta-a. Then the rate of change of TV in the direc- 
tion of a, the tangent, is represented by d a T<r. Let us 
now consider an elementary area between two neighboring 
curves of the field, and two neighboring normals. If we 
consider Ta as an intensity of some quantity whose amount 



78 VECTOR CALCULUS 

depends also upon the length of the infinitesimal normal 
curve, so that we consider the quantity Ta-dn, then the 
value of this quantity, which we will call the transport of 
the differential tube (strip in the case of a plane field), 
TV being the density of transport, will vary for different 
cross-sections of the tube, and for the case under considera- 
tion, would be Ta'dn' - Tadn. But TV' = TV + d a Ta-ds 
and dn' = dn + ds-dn times the divergence of the lines. 
Therefore, the differential of the transport will 
P" ~T~ ( be (to terms of the first order) ds X dn X ( TV 

I L— times divergence + d a Ta). Hence, the density 

F ' of this rate of change of the transport is TV 
. times the divergence + the rate of change of TV 
along the tangent of the vector line of the field. This quan- 
tity we call the divergence of the field at the initial point, and 
sometimes it will be indicated by div. cr, sometimes by 
— SVa, a notation which will be explained. It is clear 
that if the lines of a field are perpendicular to a set of straight 
lines, since the curvature of the straight lines is zero, the 
divergence of the original lines is zero, and the expression 
reduces to d a T<r. 

Curl of Field. We may also study the circulation of the 
vector a along its lines, by which we mean the product of 
the intensity TV by a differential arc, that is, Tads. On 
the neighboring vector line there is a different intensity, 
TV', and a different differential arc ds'. The differential 
of the circulation is easily found in the same manner as 
the divergence, and turns out to be 

— (dfiTa + TV X curl of the vector lines). 

This quantity we shall call the curl of the field, written 
sometimes curl a, and more frequently Wa, which notation 
will be explained. 



1 



VECTORS IN A PLANE 79 

It is evident that the curl of a is the line integral of the 
Tads around the elementary area, for the parts contributed 
by the boundary normal to the field will be zero. Hence, 
we may say that curl a is the limit of the circulation of <r 
around an elementary area constructed as above, to the 
area enclosed. We will see later that the shape of the area 
is not material. 

Likewise, the divergence is clearly the ratio to the elemen- 
tary area of the line integral of the normal component of <r 
along the path of integration. We will see that this also 
is independent of the shape of the area. 

Further, we see that in a field in which the intensity of a 
is constant the divergence becomes the geometric divergence 
times the intensity TV, and the curl becomes the geometric 
curl times the intensity T<r. 

Divergence and curl have many applications in vector 
analysis in its applications to geometry and physics. These 
appear particularly in the applications to space. A simple 
example of convergence or divergence is shown in the 
changing density of a gas moving over a plane. A simple 
caSfc of curl is shown by a needle imbedded in a moving 
viscous fluid. The angular rate of turn of the direction of 
the needle is one-half the curl of the velocity. 

13. Lines as Levels. If the general equation of a given 
set of curves is 

u(x, y) = c, 

the§e curves will be the vector lines of an infinity of fields, 
for if the differential equation of the lines is 

dx/X m dy/Y, 
then we must have 

Xdu/dx + Ydu/dy = 
and for the field 

a = Xa + Y0. 



80 VECTOR CALCULUS 

We may evidently choose X arbitrarily and then find Y 
uniquely from the equation. However, if a\ is any one 
field so determined, any other field is of the form 

a = <TiR(x, y). 

The- orthogonal set of curves would have for their finite 
equation 

v(x, y) = c 

and for their differential equation 

Xdvldy - Ydv/dx = 0. 

If we use a uniformly to represent the unit tangent of 
the u set, and P the unit tangent of the v set, then P = ha. 
The gradient of the function u is then d u-(3, and the 
gradient of the function v is — d a v-a. But the gradient 
of u is also (u x , u y ) and of v is (v x , v v ) = (u Vf — u x ). It 
follows that the tensors of the gradients are equal. In fact, 
writing Vm for gradient u, we have Vt> = kVu. We also 
have for whatever fields belong to the two sets of orthog- 
onal lines for u curves, a = rVv, for the v curves, a' = sVu, 
or also we may write 

Vv = tot, Vu = tp, a = Ta-ct. 

14. Nabla. The symbol V is called nabla, and evidently 
may be written in the form ad/dx + Pd/dy for vectors in 
a plane. We will see later that for vectors in space it 
may be written ad/dx + Pd/dy + yd/dz, where a, ft y are 
the usual unit vectors of three mutually perpendicular 
directions. However, this form of this very important 
differential operator is not at all a necessary form. In 
fact, if a and fi are any two perpendicular unit vectors in 
a plane, and dr, ds are the corresponding differential dis- 
tances in these two directions, then we have 

V = ad/dr + pd/ds. 



VECTORS IN A PLANE 81 

For instance, if functions are given in terms of r, 6, the 
usual polar coordinates, then V = Upd/dr + kUpd/rdd. 
The proof that for any orthogonal set of curves a similar 
form is possible, is left to the student. In general, V is 
defined as follows : V is a linear differentiating vector 
operator connected with the variable vector p as follows: 
Consider first, a scalar function of p, say F(p). Differentiate 
this by giving p any arbitrary differential dp. The result 
is linear in dp, and may be looked upon as the product of 
the length of dp and the projection upon the direction of 
dp of a certain vector for each direction dp. If now these 
vectors so projected can be reduced to a single vector, 
this is by definition VF. For instance, if F is the distance 
from the origin, then the differential of F in any direction 
is the projection of dr in a radial direction upon the direc- 
tion of differentiation. Hence, V7p = Up. In the case 
of plane vectors, VF will lie in the plane. In case the 
differential of F is polydromic, we define VF as a poly- 
dromic vector, which amounts to saying that a given set 
of vectors will each furnish its own differential value of dF. 
In some particular regions, or at certain points, the value 
of J7F may become indefinite in direction because the 
differentials in all directions vanish. Of course, functions 
can be defined which would require careful investigation 
as to their differentiability, but we shall not be concerned 
with such in this work, and for their adequate treatment 
reference is made to the standard works on analysis. 

We must consider next the meaning of V as applied to 
vectors. It is evident that if V is to be a linear and there- 
fore distributive operator, then such an expression as Va 
must have the same meaning as VXa. + V Y(3 + VZy if 
a = Xa + F/3 -r Zy, where a, 0, y are any independent 
constant vectors. This serves then as the definition of 



82 VECTOR CALCULUS 

Vo-, the only remaining necessary part of the definition is 
the vector part which defines the product of two vectors. 
This will be considered as we proceed. 

15. Nabla as a Complex Number. We will consider now 
p to represent the complex number x + yk, or r e , and that 
all our expressions are complex numbers. The proper 
expression for V becomes then 

V = d/dx + kd/dy = Upd/dr + kUpd/rdd. 

In general for the plane, let p depend upon two parameters 
u, v, and let 

dp = p\du -f- p 2 dv. 

If a is a function of p (generally not analytic in the usual 
sense) and thus dependent on u, v, we will have 

da = dcr/du-du + da/dv-dv = R-dpV -a. 

If we multiply dp by kpi, which is perpendicular to pi, the 
real part of both sides will be equal and we have, since kpi 
is perpendicular to pi, 

Rkpidp — dvRkpip 2 , 
and similarly 

Rkpidp = duRkpipi = — duRkpip 2 

since the imaginary part of pip 2 equals — the imaginary part 
of p 2 pi- 

Substituting in da we have 

A, = «.*,(-,*£- £+#-£) <r. 

\ Rkpip 2 ou Rkpip 2 dvJ 

The expression in (), however, is exactly what we have de- 
fined above as V, and thus we have proved that we may 
write V in the form corresponding to dp in terms of u and v : 

V = k(p 2 d/du — pid/dv)/Rkpip 2 . 

In case pi and p 2 are perpendicular the divisor evidently 



VECTORS IN A PLANE 83 

reduces to ± Tp\Tp 2 according as p 2 is negatively perpendic- 
ular to pi or positively perpendicular to it. We may write 
V in this case in the form (since p 2 = — kpi- Tp 2 /Tpi or 
+ kpr Tp 2 /Tpi) 

v = _pi_A . _Pi_ A = p f ii, p -i 1_ . 

T Pl 2 du^ T P2 2 dv F du^ Ht dv 

In any case we have dF = Rdp\/F, da = Rdp\7 -v. 

Also in any case V = Vu-d/du + \7v-d/dv. 

16. Curl, Divergence, and Nabla. Suppose now that a 
is the complex number for the unit tangent of one of a set 
of vector lines, and |8 the complex number for the unit 
tangent of the orthogonal set, at the same point. The 
curvature of the orthogonal set is the intensity of the vector 
rate of change of (3 along the orthogonal curve. But this 
is the same as the rate of change of the unit tangent a as 
we pass along the orthogonal curve from one vector line to 
an adjacent one. The differential of a is perpendicular 
to a, and hence parallel to the direction of /3. Hence this 
curvature can be written 

But if we also consider the value of R- a(R-aV)a, since the 
differential of a in the direction of a has no component 
parallel to a, this term is zero, and may be added to the 
preceding without affecting its value. Hence the curvature 
of the orthogonal set reduces to 

R(aRaV + ^/3V)« = R-Va. 

This is the divergence of the curves of a. 

If now <j = Tcr-a, we find from the definition of the 
divergence of a that it is merely 

R-Va. 
Considering in the same manner the definition of curl of a, 



84 VECTOR CALCULUS 

we find it reduces to — R-kV<r, and if we multiply this by k, 
so that we have 



curl a = - kRW(T=LV<r f 

we see at once that when added to the expression for the 
divergence of a we have 

div-<7 + curl <r = V<r. 

The real part of this expression is therefore the divergence 
of a, and the imaginary part is the curl of a. This will 
agree with expressions for curl and divergence for space of 
three dimensions. We have thus found some of the 
remarkable properties of the operator V . 

17. Solenoidal and Lamellar Vector Fields. When the 
divergence of a is everywhere zero, the field is said to be 
solenoidal. If the curl is everywhere zero, the field is called 
lamellar. 

18. Properties of the Field. Let a set of curves u = c be 
considered, and the orthogonal set v — a, and let the field a 
be expressed in the form 

o- = XVu + FVfl, 

where it is assumed that the gradients Vu, Vv exist at all 
points to be considered. We have then 

diver = RVa = RvXVu+ RvYVv _ 

+ XRWu+ YRWv. 

The expression RWu is called the plane dissipation of u. 
In case it vanishes it is evident that u satisfies Laplace's 
equation, and is therefore harmonic. 
We also have 

curl o- = I- V<r = — kRkVXVu — kRkvYVv, 

the other parts vanishing. 



VECTORS IN A PLANE 85 

Since we have chosen orthogonal sets of curves we may 
write these in the forms 

diver = (TVu) 2 dX/du + (TVv) 2 dY/dv 

+ XRVVu + YRvVv, 
curl o- = (TVu)(TW)(dY/du - dX/dv)k. 

In case we have chosen the lines of cr for the u curves, 
then X = 0, and a = Y V v 

diver = YRVW+ (TVv) 2 dY/dv, 
curl (7= TVuTVvdY/du-k. 

We notice that curl Vu = 0, curl Vv = 0, div k\/u — 0, 
divkVv = 0, kVu = VvTVu/TVv, and for 

Y = TVu/TVv, 
we have 

(TVu)- 2 RvS7u = d log (TVu/TVv)/du, 
' (fV*)~*BVV« = d log (TVv/TVu)/dv. 

We may now draw some conclusions as to the types of 
curves and <r. (Cf. B. O. Peirce, Proc. Amer. Acad. Arts 
and Sci., 38 (1903) 663-678; 39 (1903) 295-304.) 

(1) The field will be solenoidal if diver = 0, hence 

d log Y/dv = - RVW/TW 2 , ' 
which may be integrated, giving 

Y = e f(u ' v) + o{u) . 

If v is harmonic, Y is a function of u only and a =G(u)Vv. 

(2) If the field is lamellar, curl a = 0, and Y is a function 
of v only, so that a = H(v)Vv = VL(v). 

(3) If the field is both solenoidal and lamellar, 

RVVL(v) = 0, whence RVVv/(TVv) 2 = /(*), 
which is a condition on the character of the curves. Hence 



86 VECTOR CALCULUS 

it is not possible to have a solenoidal and lamellar field 
with purely arbitrary curves. 

(4) If the field is solenoidal and Ta, the intensity, 
is a function of u alone, Y = p(u)/TVv, and therefore 
d log Y/dv = - dTVv/TVvdv = - RvVv/TVv 2 , whence 

2RVW = d(TVv) 2 /dv, 

which is a condition on the curves. An example is the 
cross-section of a field of magnetic intensity inside an in- 
finitely long cylinder of revolution which carries lengthwise 
a steady current of electricity of uniform current density. 

(5) If a is lamellar and Ta is a function of v only, TVv 
= g(v). An example is the field of attraction within a 
homogeneous, infinitely long cylinder of revolution. The 
condition is a restriction on the possible curves. 

(6) If the field is lamellar and Ta a function of u only, 
since Y is a function of v only, d log TVv/du = k(u), or 
TVv = l(u)/m(v). 

This restricts the curves. 

(7) If the field is solenoidal and Ta a function of v only, 
Ta = p(v)TW. Therefore d log Ta/dv = d log TVa/dv 
— RS7Vv/(T\7v) 2 . Hence either both sides are constant 
or else both expressible in terms of v. If the field is not 
lamellar also, TVv must then be a function of u as well as 
of v. 

(8) If the field is lamellar and has a scalar potential 
function, that is, a = VP, then since a = q(v)Vv, we must 
have P a function of v only, and a = P'Vfl. From this 
it follows that diver = P\v)RVVv + P"(v)(TVv) 2 . 

(9) If the field is uniform, Ta — a, Y = a/T\7v, and a 
is lamellar only if TVv is either constant or a function of 
v only, while a is solenoidal only if we have 

2RVW = d(TVv) 2 /dv. 



VECTORS IN A PLANE 87 

(10) Whatever function u is, the u lines are vector lines 
for the vectors £ = f(u)UVv, f = g(v)U\7v, or 

T? = *(«, r)tTVf. 

(11) If the field is solenoidal, TV a function of u only, 
and the w curves are the lines of the field, then the curl 
takes the form — k div • ka, whence it has the form 

k[b(u)RVVu+ b'(u)(TVu) 2 ], 

where b may be any differentiate function. If TV is also 
a function of v, the form of the curl is 

k[b(u, v)RVVu + db(u, v)/du(TVu) 2 ]. 

(12) If TV is a function of u only, the divergence takes 
the form 

diver - Ta[RWv/TVv - dTVv/dv]. 

(13) If TV is a function of v only 

curl a = - kTaTVu/TVv-dTVv/du. 

19. Continuous Media. When the field is that of the 
velocity of a continuous medium, we have two cases to 
take into account. If the medium is incompressible it is 
called a liquid, otherwise a gas. Incompressibility means 
that the density at a point remains invariable, and if this 
is c, then from 

dc/dt= dc/dt + RaVc, = dc/dt + RV(ca) - cRV<r 
we see that the first two terms together vanish, giving the 
equation of continuity, since they give the rate per square 
centimeter at which actual material (density times area, 
since the height is constant) is changing. Hence in this case 

dc/dt = — cRV<t> 

This gives the rate of change of the density at a point 
moving with the fluid. Hence if it is incompressible, the 
velocity is solenoidal, RV& = 0. 



88 VECTOR CALCULUS 

This may also be written curl (— ka) = 0, hence — ka 
= V?, and <j — kvQ, which shows that for every liquid 
there is a function Q called the function of flow. 

When curl { = 0, we have seen that £ is called lamellar. 
It may also be called irrotational, since the curl is twice the 
angular rate of rotation of the infinitesimal parts of the 
medium, about axes perpendicular to the plane, and if 
curl { = there is no such rotation. Curl is analogous 
to density, being a density of rotation when the vector 
field is a velocity field. 

The circulation of the field is the integral fRadp along 
any path from a point A to a point B. This is the same as 
Xdx + Ydy, and is exact when 

dX/dy = dY/dx. 

But this gives exactly the condition that the curl should 
vanish. Hence if the motion is irrotational the circulation 
from one point to another is independent of the path. In 
this case we may write a = VP where P is called the 
velocity potential. 

When a is irrotational, the lines of Q have as orthogonals 
the lines of P. If the motion is rotational, these orthogonals 
are not the lines of such a function as P. If the motion is 
irrotational, we have for a liquid, RwP = 0, and P must 
be harmonic. Hence if the orthogonal curves of the Q 
curves can belong to a harmonic function they can be curves 
of a velocity potential. If a set of curves belong to the 
harmonic function u, then RWu = 0, and this shows that 
the curl of — JcVu is zero, whence Rdp(— k\/u) is exact 
= dv, where Vv = — kVu. From this we have Vm 
= kvQ for the condition that the orthogonal curves belong 
to a harmonic function. This however gives the equation 
TS/u = TvQ. We may assert then for a liquid that there 
is always a function of flow, and the curves belonging to 



VECTORS IN A PLANE 89 

this function are the vector lines of the velocity, the in- 
tensity of the velocity being the intensity of the gradient of 
the function of flow. If the orthogonal curves belong to 
a function which has a gradient of the same intensity, both 
functions are harmonic, the function of the orthogonal set 
is a velocity potential, and the motion is irrotational. 

We have a simple means of discovering the sets of curves 
that belong to harmonic functions, as is well known to 
students of the theory of functions of a complex variable, 
since the real and the imaginary part of an analytic function 
of a complex variable are harmonic for the variable co- 
ordinates of the variable. That is to say, if p = x + yk, 
and £ = /(p) = u -\- vk, then u, v are harmonic for x, y. 
The condition given by Cauchy amounts to the equation 
Vm = — k\/v, or V£ = where £ is a complex number. 
It is clear from this that the field of £ is both solenoidal and 
lamellar, a necessary and sufficient condition that £ be an 
analytic function of a complex variable. In this case £ is 
called a monogenic function of position in the plane. It is 
clear that £ = VH where H is a harmonic function. 

In case there are singularities in the field it is necessary 
to determine their effect on the integrals. For instance, if 
we have a field a and select a path in it, from A to B, or a 
loop, the flux of a through the path is the integral of the 
projection of a on the normal of the path, that is, if the path 
is a curve given by dp, so that the projection is Ra(— kdp), 
the integral of this is the flux through, the path. It is 
written 

2 = SI (- Rakdp) = - kfladp. 

In the case of a liquid the condition RV<r = shows that 
the expression is integrable over any path from A to B, 
with the same value, unless the two paths enclose a singu- 
larity of the field. In the case of a node, the integral around 

7 



90 VECTOR CALCULUS 

a loop enclosing the node is called the strength of the source 
or sink at the node. We may imagine a constant supply 
of the liquid to enter the plane or to leave it at the node, 
and be moving along the lines of the field. Such a system 
was called by Clifford a squirt 

If the circulation is taken around a singular point it 
will usually have a different value for every turn around the 
point, giving a polydromic function. These peculiarities 
must be studied carefully in each case. 

EXERCISES 

1. From £ = Ap n we find in polar coordinates that 

u = Ar n cos nd, v = Ar n sin nd. 

These functions are harmonic and their curves orthogonal. Hence 
if we set a = Vwora = V#, we shall have as the vector lines of <r the v 
curves or the u curves. What are the curves for the cases n = — 3, 
— 2, — 1, 1, 2, 3? What are the singularities? 

2. Study £ = A log p, and £ = A log (p — a)/(p + a). 

3. Consider the function given implicitly by p = £ + e*. This 
represents the flow of a liquid into or out of a narrow channel, in the 
sense that it gives the lines of flow when it is not rotational. 

4. Show that a = A/p gives a radial irrotational flow, while a = Ak/p 
gives a circular irrotational flow. What is true of a = Akpl The 
last is Clifford's Whirl. 

5. Study a flow from a source at a given point of constant strength 
to a sink at another point, of the same strength as the source. 

6. If the lines are concentric circles, and the angular velocity of any 
particle about the center is proportional to the n-th power of the radius 
of the path of the point, show that the curl is \ {n + 2) times the angular 
velocity. 

7. A point in a gas is surrounded by a small loop. Show that the 
average tangential velocity on the loop has a ratio to the average 
normal velocity which is the ratio of the tensor of the curl to the 
divergence. 

8. What is the velocity when there is a source at a fixed origin, and 
the divergence varies inversely as the w-th power of the distance from 
the origin. [The velocity potential is A log r — B{n — 2)~ 2 r 2-n .] 

9. Consider the field of two sources of equal strength. The lines are 
for irrotational motion, cassinian ovals, where, if r, r' are the distances 






VECTORS IN A PLANE 91 



from the two sources (foci) and rr' — h 2 , Q = A log h + B, the velocity 
is such that T<r = ATp/h 2 , the origin being half way between the foci; 
the orthogonal curves are given by u = iA[ir/2 — (0 + di)] where 0, 
0i are the angles between the axis and the radii from the foci, that is 
they are equilateral hyperbolas through the foci. The circulation 
about one focus is ttA, about both 2irA. 

10. If the lines are confocal ellipses given by 

z 2 /m + i/VG* - c 2 ) = 1, 

then Q = A log ( \V + V (m — c 2 )) + B. If p is the perpendicular 
from the center upon the tangent of the ellipse at any point, then the 
velocity at the point is such that T<r = — Ap/ y/ [/*(/* — c 2 )], and 
the direction of <r is the unit normal. The potential function is 
A sin -1 B' V v\c. V v is the semi-major axis. What happens at the foci? 

11. If the stream lines are the hyperbolas of the preceding, then 
a = 2 A V (*7(m — v)) times the unit normal of the hyperbola. On the 
line p = yka there is no velocity, at the foci the velocity is oo , half way 
between it is 0. The lines along the major axis outside the foci act 
like walls. 

12. If we write for brevity u x for T\7u, and vi for T\7v, show that 
we have whether the u curves are orthogonal to the v curves or not, 

V V = Ui 2 d 2 jdu 2 + Vi 2 d 2 ldv 2 + VVud/du + VVvd/dv 

+ 2RVuVvd 2 /dudv. 

If the sets of curves are orthogonal the last term vanishes; if u and v 
are harmonic the third and fourth terms drop out; if both cases happen, 
only the first two terms are left. 

13. In case of polar coordinates, Vr = Up, V0 = r" 2 A;p and 

VV = d 2 /dr 2 + r~ l dldr + r _2 d 2 /d0 2 . 

14. A gas moves in a plane in lines radiating from the origin, which 
is a source. The divergence is a function of r only, the distance from 
the center. Find the velocity and the density at any point. 

a = p f{r), flVo- = eir) = 2/(r) + rf'{r), 
and 

f(r) = Ajr 2 + r~ 2 fre{r)dr. 
To determine c, 

RV log co- = - e(r) = f(r)Rp\/ log c = rf(r)d log c/dr. 

15. Show that in the steady flow of a gas we may find an integrating 
factor for Rdpka by using the density, [dc/dt = = Rsjca = curl -fcco-, 
and Rdpkca is exact.] 

16. A fluid is in steady motion, the lines being concentric circles. 
The curl is known at each point and the tensor of a is a function of r 
only. Find the velocity and the divergence. 



92 VECTOR CALCULUS 

17. Rotational motion, that is a field which is not lamellar, is also 
called vortical motion. The points at which the curl does not vanish 
may be distributed in a continuous or a discontinuous manner. In 
fact there may be only a finite number of them, called vortices. We 
have the following: 

<r = k\7Q, VVQ = T curl a = 2«, 
Q = 7r _1 //«' log rdx'dy' + Q , 

where «' denotes co at the variable point of the integration, r is the 
variable distance from the point at which the velocity is wanted, and Q 
is any solution of Laplace's equation which satisfies the boundary 
conditions. 

If the mass is unlimited and is stationary at infinity we have 

« = kfwfftt'ifi - P ')/T(p - py-dx'dy'. 

A single vortex filament at p of strength I would give the velocity 

a =U2T.(p-p')IT(p-p')\ 

If we multiply the velocity at each point p at which there is a vortex by 
the strength, and integrate over the whole field, we find the sum is zero. 
There is then a center of vortices where the velocity is zero, something 
like a center of gravity. Instances are 

(1) A single vortex of strength I. The vortex point will remain at 
rest, and points distant from it r will move on concentric circles with 
the vortex as center, and velocity l/2wr. The circulation of any loop 
surrounding the vortex is of course the strength. 

(2) Two vortices of strengths k, U. They will rotate about the 
common center of gravity of two weighted points at the fixed distance 
apart a, the weights being the two strengths. The angular velocity of 
each is 

27ra 2 

The stream lines of the field are given by fxhf % h = const. When 
k = — {, the center is at infinity, and the vortices remain a fixed dis- 
tance apart, moving parallel to the perpendicular bisector of this segment 
joining them. Such a combination is called a vortex pair. The stream 
lines of the accompanying velocity are coaxal circles referred to the 
moving points as limit points. The plane of symmetry may be taken 
as a boundary since it is one of the stream lines, giving the motion of a 
single vortex in a field bounded by a plane, the linear velocity of the 
vortex being parallel to the wall and \ of the velocity of the liquid along 
the wall. The figure suggests the method of images which can indeed 
be applied. For further problems of the same character works on 
Hydrodynamics should be consulted. 



VECTORS IN A PLANE 93 

18. Liquid flows over an infinite plane towards a circular spot where 
it leaks out at the rate of 2 cc. per second for each cm. 2 area of the leaky 
portion. The liquid has a uniform depth of 10 cm. over the entire 
plane field. Find formulas for the velocity of the liquid inside the 
region of the leaky spot, and the region outside, and show that there is 
a potential in both regions. 

a = iVp in spot, 40/p outside, P = ^pp in spot, 40 log Tp — 20 log 
400 outside. 

Find the flux through a plane area 20 cm. long and 10 cm. high, whose 
middle line is 5 cm. from the center of the leaky spot, also when it is 
30 cm. from the leaky spot. Find the divergence in the two regions. 

Franklin, Electric Waves, pp. 307-8. 

19. Show that in an irrotational motion with sources and sinks, the 
lines of flow are the orthogonal curves of the stream lines of a correspond- 
ing field in which the sources and the sinks are replaced by vortices of 
strengths the same as that of the sources and sinks, and inversely. 
Stream lines and levels change place as to their roles. For sources and 
sinks Q = 1/2tt-ZZi0i, P = 1/2* -Z log r x h. 

20. Vector Potential. In the expression a = — VkQwe 
express (rasa vector derived by the operation of V upon 
— JcQ, the latter being a complex number. In such a case 
we may extend our terminology and call — JcQ the vector 
potential of a. A vector may be derived from more than 
one vector potential. In order that there be a vector 
potential it is necessary and sufficient that the divergence 
of <t vanish. Hence any liquid flow can have a vector 
potential, which is indeed the current function multiplied 
by — k. It is clear that Q must be harmonic. 



CHAPTER VI 

VECTORS IN SPACE 

1. Biradials. We have seen that in a plane the figure 
made up of two directed segments from a vertex enables 
us to define the ratio of the two vectors which constitute 
the sides when the figure is in some definite position. This 
ratio is common to all the figures produced by rotating the 
figure about a normal of the plane through its vertex, and 
translating it anywhere in the plane. We may also reduce 
the sides proportionately and still have the same ratio. 
The ratio is a complex number or, as we will say in general, 
a hypernumber. 

If now we consider vectors in space of three dimensions, 
we may define in precisely the same manner a set of hyper- 
numbers which are the ratios of the figures we can produce 
in an analogous manner. Such figures will be called 
biradials. To each biradial there will correspond a hyper- 
number. Besides the translation and the rotation in the 
plane of the two sides of the biradial, we shall also permit 
the figure to be transferred to any parallel plane. This 
amounts to saying that we may choose a fixed origin, and 
whatever vectors we consider in space, we may draw from 
the origin two vectors parallel and equal to the two con- 
sidered, thus forming a biradial with the origin as vertex. 
Then any such biradial will determine a single hyper- 
number. Further the hypernumbers which belong to the 
biradials which can be produced from the given biradial 
by rotating it in its plane about the vertex will be con- 
sidered as equal. 

94 



VECTORS IN SPACE 95 

The hypern umbers thus defined are extensions of those 
we have been using in the preceding chapter, the new 
feature being the different hypernumbers k which we now 
need, one new k in fact for each different plane through the 
given vertex. This gives us then a double infinity of 
hypernumbers of the complex type, r-cks 6, where the 
double infinity of k's constitute the new elements. 

2. Quaternions. The hypernumbers we have thus de- 
fined metrico-geometrically involve four essential param- 
eters in whatever way they are expressed, since the 
biradials involve two and the plane in which they lie two 
more. Hence they were named by Hamilton Quaternions. 
In order to arrive at a fuller understanding of their prop- 
erties and relations, we will study the geometric properties 
of biradials. 

In the first place if we consider any given biradial, there 
is involved in its quaternion, just as for the complex number 
in the preceding chapter, two parts, a real part and an 
imaginary part, and we can write the quaternion in the 
form 

q = r cos 6 + r sin 6 -a, 

where a corresponds to what was written k in the preceding 
chapter, and is a hypernumber determined solely by the 
plane of the biradial. On account of this we may properly 
represent a by a unit normal to the plane of the biradial, 
so taken that if the angle of the biradial is considered to be 
positive, the direction of the normal is such that a right- 
handed screw motion turning the initial vector of the 
biradial into the terminal vector in direction would in- 
volve an advance along the normal in the direction in 
which it points. It is to be understood very clearly that 
the unit vector a and the hypernumber a are distinct 
entities, one merely representing the other. The real 



96 VECTOR CALCULUS 

part of q is called, according to Hamilton's terminology, the 
scalar part of q, and written Sq. The imaginary part is 
called, on account of the representation of a as a vector, 
the vector part of q and written Vq. The unit a is called the 
unit vector of q and written UVq. The angle of q is and 
written Zq. The number r which is the ratio of the 
lengths of the sides of the biradial is called the tensor of q, 
and written Tq. The expression cos 6 + sin 6 -a = cas-d 
is called the versor of q, and written Z7^. 

Sq is a quaternion for which = 0° or 180°, Fg is a 
quaternion for which = 90° or 270°. Tq is a quaternion 
of 0°, being always positive, a is a quaternion of = 90°, 
and sometimes called a right versor. 

3. Sum of Quaternions. In order to define the sum of 
two quaternions we define the sum of two biradials first. 
This is accomplished by rotating the two biradials in their 
planes until their initial lines coincide, and then diminishing 
or magnifying the sides of one until the initial vectors are 
exactly equal and coincide. This is always possible. We 
then define as the sum of the two biradials, the biradial 
whose initial vector is the common vector of the two, and 
terminal vector is the vector sum of the two terminal 
vectors. The sum of the corresponding quaternions is 
then the quaternion of the biradial sum. Since vector 
addition is commutative, the addition of quaternions is 
commutative. 

Passing now to the scalar and vector parts of the quater- 
nions, we will prove that they can be added separately, the 
scalar parts like any numbers and the vector parts like 
vectors. 

In the figure let the biradial of q be OB/OA, of r be 
OC/OA, and of q + r be OD/OA. Let the vector part of q, 
Tq- sin Zq-UVq be laid off as a vector Vq perpendicular 



VECTORS IN SPACE 97 

to the plane of the biradial of q, and similarly for Vr. 
Then we are to show that V(q + r) = Vq + Vr in the 
representation and that this represents the vector part of 
q + r according to the definition. It is evident that 




OB = OB' + B'B, the first vector along OA, the second 
perpendicular to OA. Also OC = OC" + C"& + C'C, 
the first part along OA, the second parallel to B'B, and the 
third perpendicular to the plane of OAB. The sum 
OB + OC = OD, where OD = OB" + D"D' + D'Z), and 
0Z>" - 05' + 00", D"£' = B'B + C ,, (7 , , D'D = C'C. 

Hence the biradial of the sum is OD/OA, where the 
scalar part is the ratio of OD" to OA. This is clearly the 
sum of the scalar parts of q and r, and 

S( q + f ) = Sq+ Sr. 

The vector part of the quaternion for OD/OA is the ratio 
of D"D to OA in magnitude, and the unit part is repre- 
sented by a unit normal perpendicular to OD" and D"D. 
But D"D = B'B + C'C, and the ratio of D"D to OA equals 
the sum of the ratios of B'B and C'C to OA. If then we 
draw, in a plane through which is perpendicular to OA, 
the vector Vq along the representative unit normal of the 
plane OAB, and of a length to represent the numerical 
ratio of B'B to OA, and likewise Vr to represent the ratio 
of C'C to OA laid off along the representative unit normal 



98 VECTOR CALCULUS 

to the plane OAC, because D"D is parallel to this plane, 
as well as B'B and C"C, the representative unit vector of 
q+ r will lie in the plane, and will be in length the vector 
sum of Vq and Vr, that is V(q + r) as shown. 

It follows at once since the addition of scalars is associa- 
tive, and the addition of vectors is associative, and the two 
parts of a quaternion have no necessary precedence, that 
the addition of quaternions is associative. 

4. Product of Quaternions. To define the product of 
quaternions we likewise utilize the biradials. In this 
case however we bring the initial vector of the multiplier 
to coincide with the terminal line of the multiplicand, and 
define the product biradial as the biradial whose initial 
vector is the initial vector of the multiplicand, and the 
terminal vector is the terminal vector of the multiplier. 
In the figure, the product of the biradials OB/OA, and 




Fig. 13. 

OC/OB, is, writing the multiplier first, 

OC/OB- OB/OA = OC/OA. 

It is clear that the tensor of the product is the product of 
the tensors, so that 

T-qr= TqTr. 
It follows that 

U-qr = UqUr. 

It is evident from the figure that the angle of the product 
will be the face angle of the trihedral, AOC, or on a unit 
sphere would be represented by the side of the spherical 



VECTORS IN SPACE 99 

triangle corresponding. It is clear too that the reversal of 
the order of the multiplication will change the plane of 
the product biradial, usually, and therefore will give a 
quaternion with a different unit vector, though all the other 
numbers dependent upon the product will remain the same. 
However we can prove that multiplication of quaternions 
is associative. In this proof we may leave out the tensors 
and handle only the versors. The proof is due to Hamilton. 
To represent the biradials, since the vectors are all taken 
as unit vectors, we draw only an arc on the unit sphere, 
from one point to the other, of the two ends of the two unit 
vectors of the biradial. Thus we represent the biradial 
of q by CA, or, since the biradial may be rotated in its 
plane about the vertex, equally by ED. The others in- 
volved are shown. The product qr is represented by FD, 
from the definition, or equally by LM. What we have 
to prove is that the product p • qr is the same as the product 
pq-r, that is, we must prove that the arcs KG and LN are 
on the same great circle and of equal length and direction. 




Fig. 14. 

Since FE = KH, ED - CA, HG = CB, LM = FD, the 
points L, C, G, D are on a spherical conic, whose cyclic 
planes are those of AB, FE, and hence KG passes through 
L, and with LM intercepts on AB an arc equal to AB. 
That is, it passes through N, or KG and LN are arcs of the 



100 VECTOR CALCULUS 

same great circle, and they are equal, for G and L are points 
in the spherical conic. 

5. Trirectangular Biradials. A particular pair of bira- 
dials which lead to an interesting product is a pair of which 
the vectors of each biradial are perpendicular unit vectors, 
and the initial vector of one is the terminal of the other, 
for in such case, the product is a biradial of the same kind. 
In fact the three lines of the three biradials form a tri- 
rectangular trihedral. If the quaternions of the three 




o 

Fig. 15. 

are i, j, k, then we see easily that the quaternion of the 
biradial OC/OB is represented completely by the unit vector 
marked i, the quaternion of OA/OC by j, and of OB/OA by 
k. The products are very interesting, for we have 

ij = k, jk = i, hi = j, 

and if we place the equal biradials in the figure we also have 

ji = — k, kj = — i, ik = — j. 

Furthermore, we also can see easily that, utilizing the 
common notation of powers, 

V- = - 1, ? - - 1, V - ■■- 1. 

Since it is evidently possible to resolve the vector part of 
any quaternion, when it is laid off on the unit vector of its 
plane as a length, into three components along the direc- 
tions of i, j, k, and since the sum of the vector parts of 



VECTORS IN SPACE 



10] 



quaternions has been shown to be the vector part of the 
sum, it follows that any quaternion can be resolved into 
the parts 

q = w -\- xi -\- yj -\- zk. 

These hypernumbers can easily be made the base of the 
whole system of quaternions, and it is one of the many 
methods of deriving them. Hamilton started from these. 
The account of his invention is contained in a letter to a 
friend, which should be consulted. (Philosophical Maga- 
zine, 1844, vol. 104, ser. 3, vol. 25, p. 489.) 

6. Product of Vectors. It becomes evident at once if we 
consider the product of two vector parts of quaternions, 
or two quaternions whose scalar parts are zero, that we 
may consider this product, a quaternion, as the product of 
the vector lines which represent the vector parts of the 
quaternion factors. From this point of view we ignore 
the biradials completely, and look upon every geometric 
vector as the representative of the vector part of a set of 
quaternions with different scalars, among which one has 
zero scalar. From the biradial definition we have 

VqVr= S-VqVr+ V-VqVr 

equal to the quaternion whose biradial consists of two 
vectors in the same plane as the vector normals of the 




Fig. 16. 



102 VECTOR CALCULUS 

biradials of Vq, Vr and perpendicular to them respectively. 
In the figure the biradial of Vr is OAB, and of Vq is OBC, 
and of VqVr is OAC. If then we represent the vectors by 
Greek letters whether meant to be considered as lines or 
as vector quaternions, a = Vq, /3 = Vr, then the quaternion 
which is the product of a(3 has for its angle the angle be- 
tween /3 and a + 180°, and for its normal the direction OB. 
If we take UVa(3 in the opposite direction to OB, and of 
unit length, so as to be a positive normal for the biradial 
a /3 in that order, then we shall have, letting 6 be the angle 
from a to /3, 

a(3 = TaTj3(- cos + UVafi sin 0). 

We can write at once then the fundamental formulae 

S-a& = - TaTfi cos 6, V-a$ = TaTp-sm 6- UVaP. 

From this form it is clear also that any quaternion can 
be expressed as the product of two vectors, the angle of 
the two being the supplement of that of the quaternion, 
the product of their lengths being the tensor of the quater- 
nion, and their plane having the unit vector of the quater- 
nion as positive normal. 

If now we consider the two vectors a and to be resolved 
in the forms 

a = ai-\- bj + ck, (3 — li + mj + nk, 

where i, j, k have the significance of three mutually tri- 
rectangular unit vectors, as above, then since Ta Tfi cos 6 
= al-\- bm-\- en, and since the vector Ta T(3 sin 6 • UVa(3 
is 

(bn — cm)i + (cl — an)j + (am — bl)k, 
we have 

a/3 = — (al + bm + en) + (bn — cm)i 

+ (cl — ari)j -\- (am — bl)k. 



VECTORS IN SPACE 103 

But if we multiply out the two expressions for a and 
distributively, the nine terms reduce to precisely these. 
Hence we have shown that the multiplication of vectors, 
and therefore of quaternions in general, is distributive when 
they are expressed in terms of these trirectangular systems. 
It is easy to see however that this leads at once to the 
general distributivity of all multiplications of sums. 

7. Laws of Quaternions. We see then that the addition 
and multiplication of quaternions is associative, that 
addition is commutative, and that multiplication is dis- 
tributive over addition. Multiplication is usually not 
commutative. We have yet to define division, but if 
we now consider a biradial as not being geometric but as 
being a quaternion quotient of two vectors, we find that 
P/a differs from a(3 only in having its scalar of opposite 
sign, and its tensor is T(3/Ta instead of TaTfi. 

It is to be noticed that while we arrived at the hyper- 
numbers called quaternions by the use of biradials, they 
could have been found some other way, and in fact were so 
first found by Hamilton, whose original papers should be 
consulted. Further the use of vectors as certain kinds of 
quaternions is exactly analogous, or may be considered to 
be an extension of, the method of using complex numbers 
instead of vectors in a plane. In the plane the vectors 
are the product of some unit vector chosen for all the plane, 
by the complex number. In space a vector is the product 
of a unit vector (which would have to be drawn in the 
fourth dimension to be a complete extension of the plane) 
by the hypernumber we call a vector. However, the use of 
the unit in the plane was seldom required, and likewise in 
space we need never refer to the unit 1, from which t^e 
vectors of space are derived. On the other hand, just as 
in the plane all complex numbers can be found as the ratios 



104 VECTOR CALCULUS 

of vectors in the plane in an infinity of ways, so all quater- 
nions can be found as the ratios of vectors in space. All 
vectors are thus as quaternions the ratios of perpendicular 
vectors in space. And multiplication is always of vectors as 
quaternions and not as geometric entities. In the common 
vector systems other than Quaternions, the scalar part of 
the quaternion product, usually with the opposite sign, 
and the vector part of the quaternion product, are looked 
upon as products formed directly from geometric con- 
siderations. In such case the vector product is usually 
defined to be a vector in the geometric sense, perpendicular 
to the two given vectors. Therefore it is a function of 
the two vectors and is not a number or hypernumber at 
all. In these systems, the scalar is a common number, and 
of course the sum of a number and a geometric vector 
is an impossibility. It seems clear that the only defensible 
logical ground for these different investigations is that of 
the hypernumber. 

It is to be noticed too that Quaternions is peculiarly 
applicable to space of three dimensions, because of the 
duality existing between planes and their normals. In a 
space of four dimensions, for instance, a plane, that is a 
linear extension dependent upon two parameters, has a 
similar figure of two dimensions as normal. Hence, corre- 
sponding to a biradial we should not have a vector. To 
reach the extension of quaternions it would be necessary 
to define triradials, and the hypernumbers corresponding 
to them. Quaternions however can be applied to four 
dimensional space in a different manner, and leads to a 
very simple geometric algebra for four-dimensional space. 
The products of quaternions however are in that case not 
sufficient to express all the necessary geometrical entities, 
and recourse must be had to other functions of quaternions. 



VECTORS IN SPACE 105 

In three-dimensional space, however, all the necessary ex- 
pressions that arise in geometry or physics are easily 
found. And quaternions has the great advantage over 
other systems that it is associative, and that division is 
one of its processes. In fact it is the most complex system 
of numbers in which we always have from PQ = the 
conclusion P = 0, or Q = 0.* 

8. Formulae. It is clear that if we reverse the order of 
the product ce/3 we have 

0a = Soft - Vafi. 

This is called the conjugate of the quaternion a(3, and 
written K-a(3. We see that 

SKq = Sq= KSq, VKq = - Vq = KVq. 

Further, since 

qr = SqSr + SqVr + SrVq + VqVr, 
we have 

K-qr= SqSr - SqVr - SrVq + VrVq = KrKq. 

From this important formula many others flow. We have 
at once 

K-qi- • -q n = Kq n > • >Kqi. 
And for vectors 

Koli- • -0L n = {—) n a n - • •«!. 
Since 

Sq = i(q+Kq), Vq=\{q-Kq), 

we have therefore 

S-OLl" 'Qt2n = i(<*l" * -«2n + «2n ' * 'Oil), 

S-ai- • -C^n-l = i(tti* • 'tt2n~l — «2n-l' * 'Oil), 

V'CXi- ' 'OL 2n = !(«!« * ,Q; 2n ~ « 2n ' " '«t), 

F'Qfi- • ■a 2 n-l = %(<Xi' ' -OL2n-\ + «2n-l ' * -«l). 

* Consult Dickson: Linear Algebras, p. 11. 



106 VECTOR CALCULUS 

In particular 

2Sa$ = aft + Pa, 2SaPy = afiy - y(3a, 
2Vap = a/3 - fa 2Va(3y = a(3y + y(3a. 

It should be noted that these formulae show us that both 
the scalar and the vector parts of the product can them- 
selves always be reduced to combinations of products. 

This is simply a statement again of the fact that in 
quaternions we have'only'one kind of multiplication, which 
is distributive and associative. 

We see from the expanded form above for S • qr that 

S-qr = S-rq. 

Hence, in any scalar part of a product, the factors may be 
permuted cyclically. For instance, 

S-afi = S-(3a, S-a(3y = S-Pya = S-yaQ, 
S-a(3y5 = SPyfa 

From the form of 

Sq=Uq+Kq), Sq = SKq; 

hence we have 

Sa(3 = S@a, Safiy = - Syfa Sa(3y8 = S8y(3a, etc. 

From the form of VKq = — Vq we see that 

Vafi = - V@a, Vafiy = VyPa, 
Vapyh m - Vdypa, Vapyhe = VebyPa- 

We do not have a simple relation between V-qr and 
V-rq, but we have the fact that they are respectively the 
sum and the difference of two vectors, namely, 

If a — SqVr -+- SrVq, P = VVqVr, then ft is perpendicular 
to a, and 

Vqr = a + P, Vrq = a — (3. 



q = w + 


w? + , 


;V + ^ 


(Tq) 2 = w 2 + 


x 2 + 


2/ 2 + * 2 , 


£g = 


», 




(TTg) 2 - 


z 2 + 


f+z\ 



VECTORS IN SPACE 107 

It is obvious that TVqr = TVrq and that /.qr = /rq 
- tan -1 TVqr/Sqr. The planes differ. 

The product of g and i£g is the square of the tensor of q. 
We indicate the unitary part of q, called the versor of q, 
by Uq. We have then the formulae 

Kq = w — ix — jy — kz, 
j j = w + ix + jy + kz 
q Tq 

Vq = ix + jy + kz, 

TTVn _ix + jy + fa 
^ Kg rrg ' 

(TVUq) 2 m (X* + f + * 2 )/(w 2 + a? + 2/ 2 + z 2 ), 
cos- Z g = w/Tg = #• £7g, 
sin-Z g = TVq/Tq= TVUq, 
Z-q= tan" 1 rFg/.Sg. 

The product of two quaternions is 

qr = ww* — xx' — yy' — zz' -f i(wx' -\- w'x + yz f — y'z) 

+ j(wy' + w'y + zx' — z'x) 

+ k(wz' + w'a + xy f — x'y). 

From the formula Tqr = TgTY we have a noted identity 

(ief +a*+ y 2 + z 2 ) <>' 2 + x' 2 +y' 2 + s' 2 ) 
= (ww f — ao' — 2/2/' — zz') 2 + (wa;' + w'x + 2/2' — S/'s) 2 
+ (wy' + to'y + zx' — z'x) 2 + (W + w'z + #2/' — ^'2/) 2 - 

This formula expresses the sum of four squares as the 
product of the sums of four squares. It was first given by 
Euler. The problem of expressing the sum of three squares 
as the product of sums of three or four squares and the 
sum of eight squares as the product of sums of eight squares 
has also been considered. 



108 



VECTOR CALCULUS 




9. Rotations. We see from the adjacent figure that we 
have for the product 

qrq- 1 

a quaternion of tensor and angle the same as that of r. 
But the plane of the product is produced by rotating the 
plane of r about the axis of q through an angle double the 
angle of q. In case r is a vector /3 we have as the product 
a vector fi f which is to be found by rotating conically the 
vector (3 about the axis of q through double the angle of q. 
It is obvious that operators* of the type qQq~ l , r()r -1 , 
which are called rotators, follow the same laws of multiplica- 
tion as quaternions, since g(r()r _1 )<7 -1 = qrQ[qr]~ l . A 
gaussian operator is a rotator multiplied by a numerical 
multiplier, and is called a mutation. The sum of two 
mutations is not a mutation. As a simple case of rotator 
we see that if q reduces to a vector a we have as the result 
of after 1 = /3' the vector which is the reflection of /3 in a. 
The reflection of /3 in the plane normal to a is evidently 

— a$or l . 

EXAMPLES 
(1) Successive reflection in two plane mirrors is equivalent 
* QOq' 1 represents a positive orthogonal substitution. 



VECTORS IN SPACE 109 

to a rotation about their line of intersection of double their 
angle. 

(2) Successive reflection in a series of mirrors all per- 
pendicular to a common plane, 2h in number, making 
angles in succession (exterior) of <pu, (P23, <&*••• is equivalent 
to a rotation about the normal to the given plane to which 
all are orthogonal, through an angle 6 = 2h — ir — 2(<p 12 
+ (pu + ••• + <P2h-i,2h) which is independent of the 
alternate angles. 

(3) Study the case of successive reflections in mirrors in 
space at any angles. 

(4) The types of crystals found in nature and possible 
under the laws that are found to be true of crystals, are 
solids such that every face may be produced from a single 
given face, so far as the angles are concerned, by the 
following op9rations : 

I, the reversal of a vector, in quaternion 

form — 1 . 

A, rotation about an axis a a n ()oT n . 

L4, rotatory inversion about a — a n ()a~ u . 

S, reflection in a plane normal to /5 — jSO/S -1 = /?()/?. 

The 32 types of crystals are then generated by the succes- 
sive combinations of these operations as follows: 

Triclinic Ci Asymmetric 1. 

d Centre-symmetric 1,-1. 

Monoclinic C s Equatorial 1, 0Q0. 

d Digonal polar 1, a()a -1 . 

C 2 h Digonal equatorial 1, a()a; -1 , a()a. 

Orthorhombic C 2v Didigonal polar 1, a()a~ l , 0Q0, Sap = 0. 

D 2 Digonal holoaxial 1, a()a-\ fiQfi' 1 , Sap = 0. 

Du Didigonal equatorial .... 1, a()a -1 , POP' 1 , «()«, 

SaP = 0, 
A = a l ' 2 0a- 1 ' 2 . 
Tetragonal d Tetragonal alternating . .1, — A. 

Du Ditetragonal alternating. 1, — A, P{)P~ X . 

d Tetragonal polar 1, A. 



110 VECTOR CALCULUS 

Ctk Tetragonal equatorial. . .1, A, aQa. 

C4* Ditetragonal polar 1, A, /3()/3. 

D4 Tetragonal holoaxial .... 1, A, 0Q0~K 

Dak Dietragonal equatorial . . 1, A, aQa, /3()/3 _1 . 
Rhombohedral C 8 Trigonal polar l,B, where B is a 2l3 0<*~ il3 ' 

Czi Hexagonal alternating . .1, B, — B. 

Ctv Ditrigonal polar 1, B, pQ0. • 

D, Trigonal holoaxial 1, B, 0Q0T+. 

Did Dihexagonal alternating . 1, B, j8()/8~ l , 7O7, 7 

bisects Z/3, B0. 
Hexagonal Czh Trigonal equatorial 1,5, aQa. 

Dzh Ditrigonal equatorial . . .1, B, aQa, jS()/3 _1 . 

d Hexagonal polar 1, C, where C = a 1/3 ()«~ 1/3 . 

dh Hexagonal equatorial . . . 1, C, aQa. 

Civ Dihexagonal polar 1, C, /3()yS, where Sap = 0, 

bisects angle of 7 and 
Cy, Say = 0. 

Di Hexagonal holoaxial .... 1, C, /3()/S _1 . 

Dan Dihexagonal equatorial. .1, C, a()a, pQ(3~ l . 

Regular T Tesseral polar ..1, aQa' 1 , PQP~ X , Safi 

= Spy = Sya = 0, L 
where L = (a + fj 

+ 7)0(«+/3 + 7 )- 1 . 

T h Tesseral central 1, aQa~\ 0Q/T 1 , 7O7" 1 , 

L, aQa. 
T d Ditesseral polar 1, aQa' 1 , 0Q0-\ 7O7" 1 , 

L, (a + fi)Q(a + /3). 
Tesseral holoaxial 1, aQa-\ 0Q0~ l t yQy~ l , 

L, (a + p)Q(a + P)~K 
Oh Ditesseral central 1, aQa~\ $00-*, yQy' 1 , 

t, t {« + 0)Q(a+0? t 

aQa. 

The student should work out in each case the fuJl set of 
operators and locate vectors to equivalent points in the 
various faces. 

Ref. — Hilton, Mathematical Crystallography, Chap. IV- 
VIII. 

(5) Spherical Astronomy. We have the following nota- 
tion: 

X is a unit vector along the polar axis of the earth, 
h is the hour-angle of the meridian, 



VECTORS IN SPACE 111 

L = cos h/2 + X sin h/2, 
i = unit vector to zenith, 
j = unit vector to south, 
k = unit vector to east, X = i sin I — j cos /, where I is 

latitude, 
li = unit vector to intersection of equator and meridian, 

\x — i cos I -\- j sin I, aSX/x = SkX = Sk/j, = 0, 
d = declination of star, 

5 = unit vector to star on the meridian = X sin d + jjl cos d, 
z = azimuth, 
A = altitude. 
At the hour-angle h, 8 becomes 8' = L~ l 8L. 

The vertical plane through 8 f cuts the horizon in 
iVi8' = JSJ8' + kSk8', tan z = Sk8'/Sj8'. 
At rising or setting z is found from the condition Sid' = 0. 
The prime vertical circle is through i and k. The 6-hour 
circle is through X and V\ji. 
a — right ascension angle, 
t = sidereal time in degrees, 
h = t + a, 

L t = cos t/2 + X sin t/2, 
L a = cos a/2 + X sin a/2, 
e = pole of ecliptic, 

X = first point of aries = vernal equinox = Lr l ^L t} 
s = longitude, 
b = latitude, 
M = cos s/2 + e sin s/2. 

Problems. Given /, d, find A and z on 6-hour circle. 
Sfx8' = 0. 
/, d, find h and z on horizon. 
/, d, find A. 

I, d, A, find h and z, 8' = L~ l 8L = i 
cos A + ? cos s + k sin 2. 



112 VECTOR CALCULUS 

/, d, h, find A and z. 
a and d, find s and b. 

(G) The laws of refraction of light from a medium of 
index n into a medium of index n' are given by the equation 

nVvct — n'Vva! 

where v, a, a' are unit vectors along the normal, the 
incident, and the refracted ray. 
The student should show that 

Investigate two successive refractions, particularly back 
into the first medium. 

(7) It is easy to show that if q and r are any two quater- 
nions, and /3 = V • VqVr, we may write 

(8) For any two quaternions 

qiq' 1 ± r _1 ) = (r =b q)f\ and = r(r ± q)~ l q. 

-± - 

9 r 

(9) If a, b, c are given quaternions we can find a quater- 
nion q that will give three vectors when multiplied by a, b, 
c resp. That is, we can find q, a, ft y such that 

aq = a, bq = ft eg = 7. (R. Russell.) 

We have a — — V • Vc/aVa/b, etc., or multiples of these. 

(10) In a letter of Tait to Cayley, he gives the following: 

(q+ r)()(g+ r)" 1 = (qlr) x rQf-i( q /r)-* 

= qiq-iryQiq-^-vq- 1 = qh^Qq-^q- 1 , 
(Vq+ Vr)()(Vq+ Fr)" 1 = fa/rWJf^fo/r)- 1 /*, 



VECTORS IN SPACE 113 

where tan xA = a sin A/ (a cos A + 1), c sin 2la sin ra/3 
+ cos 2la cos ra/3 = 2 (a cos o + & cos /S) V (6 sin |8), 
2c + sin 2/a cos ra/3 = 2a sin a/ (6 sin /3). 

Interpret these formulae. 

10. Products of Several Quaternions. We will develop 
some useful formulae from the preceding. 

If we multiply a(3-(3a we have 

a 2 (3 2 - S 2 a(3 - V 2 a(3. 

Since Sax = 0, if x is a scalar, 

&*/3t = SaVfry, Sa(3y8 = SaVfiyb, etc. 
Since 

2Va(3 = a(3 - (3a, 2Sa(3 = a(3 + 0ce, 
ffiaV(3y = af3y — ay (3 — (3ya + 7/fa = 2(7/3o — 07/?) 

= 2(y(3a + 7«/3 — ay {3 — yap). 
For 

2<S/?7 • a = /57a: + 7/fo = 2aSj3y = 0:187 + 0:7/?, 
whence 

0:187 — $70 = Yj8o — 07/?. 
Therefore 

VaV(3y = ySa(3 - (3Say. 

Adding to each side ccSfiy, we have 

Va(3y = aS(3y - (3Sya + ySa(3. 
Since 

]S = crtaft = a^SaP + a~Wa$, 

which resolves (3 along and perpendicular to a, 

Sqrq -1 = Sr = qSrq -1 , 

Vqrq- 1 = h^q~ l - Kq~ l KrKq) 

= iC^a -1 — qKrq~ l ) = qVr-q~ l . 

That is, if we rotate the field, Sr and TTr are invariant. 



114 VECTOR CALCULUS 

Hence Vapy = VafiyaoT 1 = aV(3ya-oT l and Vafty, 
Vfiya. are in a plane with a and make equal angles with a. 
For instance if a, /?, y, Vafly, Vfiya, Vyafi intersect a 
sphere, then a, /?, y bisect the sides of the triangle Vafiy, 
Vpya, Vya(3, a being opposite to Vya(3, etc. Evidently if 
«i, (X2- • -a n are n radii of a sphere forming a polygon, then 
they bisect the sides of the polygon, given by Vaia 2 - • -a n , 
F«2«3- • '<x n , Vets- - -a n aia2, • • -Va n (xi- - -a n -i. This ex- 
plains the geometrical significance of these vectors. In 
fact for any vector a and quaternion q, the vector a bisects 
the angle between Vqa and Vaq, that is to say we construct 
Vqa from the vector Vaq by reflecting it in a. The same 
is true for any product, thus (3yde • • • vol is different from 
a(3y8e • • • v only in the fact that its axis is the reflection in 
a of the axis of the latter. 

<M3 ' ' ' Qnqi differs from qiq 2 • • • q n only in the fact that 
its axis has been rotated negatively about the axis of q\ 
through double the angle of qi. Indeed 

?2?3- • -q n qi = q~Kqiq2- ■ -qn)q\. 

If we apply the formula for expanding VaVfiy to 
V(Vafi)Vy8 = — V(Vy8)Va(3 we arrive at a most im- 
portant identity: 

V-VapVy8 = 8Sa$y - ySa$8 

= - V-VydVafi = aS(3y8 - /3Say8. 

From this equality we see that for any four vectors 

8Sapy = aSfiyd + @Sya8 + ySaj38. 

This formula enables us to expand any vector in terms of 
any three non-coplanar vectors. Again 
5Sapy - VpySad = V-aV(V(3y)8 

= - V-aV8V$y = Fa(3Sy8 - VayS(38. 



VECTORS IN SPACE 115 

We have thus another important formula 

SSofiy = Va(3Sy5 + VfiySaB + VyaS08, 

enabling us to expand any vector in terms of the three 
normals to the three planes determined by a set of three 
vectors, that is, in terms of its normal projections. Since 

aSPyS = VpySad + VytSefi + VbfiSay 

and 

(3Syda = Vay S{38 + VySSofi + VdaSPy, 
we have 

VVapVyd = VabSPy + VPySad - VayS(38 - VpbSay. 

From this we have at once an expansion for Vafiyh, namely 

Vctfyd = Va(3Sy8 - VaySpb + VabSPy 

+ SapVyb - SayVpb + SabVpy. 
Also easily 

Sapyd = SaPSyd - SaySpd + SabSpy. 
SVapVyb = SadSPy - SaySpb. 
V-ap-Sybe = yS-VapVbe - bS-VapVye + eS- VapVyb 

y b e 

Say Sab Sae 

SPy SPB Spe 

In the figure the various points lie on a sphere of radius I. 
The vectors from the center will be designated by the 
corresponding Greek letters. The points X, Y, Z are the 
midpoints of the sides of the A ABC. From the figure it 
is evident that 

H» = yli = (7/« 1/2 , v/y - «h m (a/7) 1 *, 
Whence 

7 = sar 1 , « - nrr\ p = ^r 1 , 



116 



VECTOR CALCULUS 




P 
Fig. 18. 
where 

v = it 1 !, 

and the axis of p is ± a. Also p%p~ l = ^iT 1 ^ 7 ? -1 * so that 
if P is the pole of the great circle through XY then the 
rotation pQp~ l brings £ to the same position as the rotation 
around OP through twice the angle of tjJ -1 . Since £ goes 
into {' by a rotation about OA as well as one about 0P f 
this means that the new position 0Z r is the reflection of OZ 
in the plane of OP A. The angle of p is then ZAL or ZAP 
according as the axis is -\- a or — a. The angles of L and 
M are right angles, and if we draw CN perpendicular to 
XY then 



ANCY = ALAY, ANCX = AMBX, 



and 



AL = BM = CN and APB is isosceles. 

Hence the equal exterior angles at A and B are ZAL 
= ZBM = \{A + 5 + Q. 

Draw PZ, then /ZiM = Zv^ 1 for it =JzJWM 
= \ML = ZF since ilfZ - XN and iVF = YL. The 
angle between the planes LAP and ZOP is thus the biradial 
7)%~ l and also £" is the biradial whose angle is that of the 



VECTORS IN SPACE 117 

planes OAZ, ZOP, so that ZOA and AOL make an angle 
equal to z p, hence 

ZV = h(A + B+C). 
Further 

pa' 1 - nlyyfc'tla = («/t) 1/2 (t/« 1/2 (/3/«) 1/2 - p'. 

The angle of p' is thus %(A + 5 + C - tt) = 2/2 where S 
is the spherical excess of AABC. 

Consider the quaternion p = r)^ 1 ^ = — 77^". The con- 
jugate of p is Kp = ££77, whose axis is also a and angle 

- \{A + B + 0). Thus the quaternion ffij = - sin 2/2 

- a: cos 2/2. 

Shifting the notation to a more symmetric form we have 
for any three vectors 

aia 2 as = — sin 2/2 — TJVai(x 2 a.z • cos 2/2 

= cos \<j — k sin Jo - , 

where 2 is the spherical excess of the triangle the midpoints 
of whose sides are A\, A 2 , A% and a is the sum of the angles 
of the triangle. Hence 

Saia 2 a 3 = cos Jo", Va ia 2 a 3 = ~* UV<x\ol 2 ccz sin \a. 

It is to be noted that the order as written here is for a 
positive or left-handed cycle from A\ to A 2 and A$. Since 
2 is the solid angle of the triangle, — S-a\a 2 as is the sine 
of half the solid angle and — TVa\a 2 az is the cosine of half 
the solid angle, made by oi, a 2 , a 3 . 

If now we have several points as the middle points of the 
sides of a spherical polygon, say aia 2 - • -a n and the vertex 
between a\ and a n is taken as an origin for spherical arcs 
drawn as diagonals to the vertices of the polygon, then for 
the various successive triangles if we call the midpoints of 
the successive diagonals 

J*lj $2, ' ' "fn-3 



118 VECTOR CALCULUS 

we have, taking the axis to the origin which we will call k, 
and which is the common axis of all the quaternions made 
up by the products of three vectors 

The sum of the angles of the polygon is the sum of the 
angles of all the triangles into which it is divided, so that 
if this sum is a we have for any spherical polygon 

«i«2- • *«n = (— ) n_3 [cos cr/2 — k sin a/2]. 

We are able to say then that if the midpoints of the sides 
of a spherical polygon are ai, a 2 , • • -a nt then 

SoCi(X2' ' '0i n = db COS ff/2, 

where a is the sum of the angles ; the vertices of the polygon 
are given by 

Wolioli- • -a n , TJVcioOLz - • • a n ai, ••'•, 
UVa n - • -ttn-l, 

each being the vertex whose sides contain the first and last 
vectors in the product; and the tensors of these vectors are 
each equal to sin <r/2. 

The expression — Sa(3y is called the first staudtian of 
afiy, the second staudtian is 

- SVapVPyVya/TVapTVPyTVya 

= S 2 aj3y/TVaPTV(3yTVya, 

which is evidently the staudtian of the polar triangle. 

S-ai--a n ,i r i i 
mrz — — — • = tan f solid angle. 
1 V •«!• • -a n 

We will summarize here the significance of the expressions 
worked out thus far, and in particular the meaning of their 
vanishing. 



VECTORS IN SPACE 119 

Sa(3 is the product of TaTp by the cosine of the angle 
between a and — 0. It vanishes only if they are per- 
pendicular. 

Vafi is the vector at right angles to both a (3 whose length is 
TaTfi multiplied by the sine of their angle. It vanishes 
only if they are parallel. 

Safiy is the volume of the parallelepiped of a fi y, taken 
negatively. It vanishes only if they are all parallel to 
one plane. 

Vafiy, Vafiyd, • •'• these vectors are the edges of the poly- 
hedral giving the circumscribed polygon, and if the ex- 
pression vanishes, we have by separating the quaternion, 

Va0y8- • • = aS(3y8- • • + VaVPyS-'- = 0. 

Hence a is the axis of (3yd- • • and Sfiyd- • • equals zero. 
By changing the vectors cyclically we have n vectors 
all of which have a zero tensor, so that each edge is the 
axis of the quaternion of the other n — 1 taken cyclically. 
This quaternion in each case has a vanishing scalar. 
n = 3, a j8 y are a trirectangular system. 
n = 4, a (3 y 8 are coplanar, shown by the four vanish- 
ing scalars. The angle a(3 = angle 7#. 
n = 5, the edge Va(3y is parallel to V8e and cyclically 

similar parallelisms hold. 
We have in all these cases the sum of the angles of the 
circumscribing polygon a multiple of 2w and it 
satisfies the inequality S(n — 2)tt is greater than 
a which is greater than {n — 2)x. It is evident 
that if the polygon circumscribed has 540° the 
vectors lie in one plane. ■ 
Safiyb = 0. If e = Va(3y8, then VaQySe = 0, and the 

preceding case is at hand for the five vectors. 
S-aia 2 - • -oL n = 0, the sum of the angles of the polygon is 
an odd multiple of x. 



120 



VECTOR CALCULUS 



EXERCISES 

1. S-VaPVpyVya = - (Sapy)* 
V-VapVpyVya = VaP(y*SaP - SPySya) + ..... 

2. S(a + P)iP + 7)(7 + «) m 2Sa0y. 

3. 5-F(a + /3)(0 + 7)708 + 7)(7 + a)V(y + «)(a + 0) 

4. 5.F(Fa/3F/37)(F/37^7«)7(F7aFa/3) = - (S-afiy)*. 

5. S-5ef - - 16(5 -a^) 4 , 
where 

5 = F(F[« + 0[\fi + 7]F[^ + 7 ][7 + «]), 
< = y(7D9 + 7][7 + a]V[y + a][a + fl), 
f = V(V[y + «][a + /S]7[a + 0]\fi + 7]). 

6. S(xa + yP + 27 )(x'a + y'0 + *'7)(x"a + y"0 + *"7) 



4(5.afl 7 ) 1 . 



7. 











x \ 






X' 
X" 


S-aiPiyi = 


- 


Saai Sftai Syai 
Sa0i S00i Syfii 
Say 1 Sfiyi Syyi 


Saai Sa&i 

S0 ai sm 

Syai Syffi 
S8ai S8P1 


s 
s 
s 

,8 


ayi Sadi 
Pyi SP81 
771 Sy8i 
571 S881 




■■ 



S • a/37. 



for any eight vectors. If the element Saai is changed to Szai the value 
is - S-0y8'S'Piyi8i-S-ai(e — a). 

9. S-Va0yV0yaVyaP = ISaPSPySyaSaPy. 

10. From S 2 P /a - V 2 P fp = 1 we find 



T(Sp/a + Vp/P) 
where 



1 = T{\cl+ p + \pa~i 



- irv + yr 1 ) = T(a' P + p/80 

a' = §(«T* - r>), pV = i(a"» + p*-«). 

11. If T P = Ta = Tp = 1 and S-afip = 0, 

S-U(p-a)U(p -P) = ±iV[2(l -Sap)]. 

12. If a, P, 7 and a h Pi, 71 are two sets of trirectangular unit vectors 
such that if a = Py, a x = Piy, then we may find angles called Eulerian 
angles such that 

a 2 = a COS yp + P sin \J/, P 2 = — a sin i£ + P COS ^, 

73 = 7 cos 6 + <*2 sin 0, a 3 = — 7 sin -f « 2 cos 0, 

71 = 73, «i = «3 cos ^ + /?2 sin <p, 

Pi = — a 3 sin v> + /3 2 cos <p. 



VECTORS IN SPACE 121 

13. If q = ai«2 • • • ot n then if we reflect an arbitrary vector in 
succession in a„, a n -i, • • • 0:20:1 when Sq = the final position will be a 
simple reflection of p in a fixed vector, and if Vq = the final position 
will be on the line of p itself. Similar statements hold if the reflections 
are in planes that are normal respectively to a n , • • • «i. 

11. Functions. We notice some expressions now of the 
nature of functions of a quaternion. We have the follow- 
ing identity which is useful : 

(a/3) n + {$a) n = (ol$ + $a)l(<xP) n ~ l ] ~ a^a[(a^ n ~ 2 

= 2SaP[(a(3) n ~ 1 + (/to) 71-1 ] - a 2 ^[(a^ n ~ 2 

+ 08*)*-*]. 

Whence 2 n S n a(3 = (a/3 + M n = [(«/3) n + (fax)"] 

+ lt/ nl ni K«/5)"- 2 + (/3a)"" 2 ]a 2 /3 2 
\\{n — 1)1 

+ 2l(n w l 2)1 [(«» n_4 + w-v/3 4 + • • • 

\\{n — 1)1 

This implies the familiar formula for the expansion of cos n 
in terms of cos nd, cos (n — 2)0, and we can write as the 
reverse formula 

S(a(3) n - (-) w / 2 [a n /3 n - n 2 S 2 a(3-a n - 2 l3 n - 2 l2\ 

+ n 2 (n 2 - 2 2 )SV-a n " 4 /3"- 4 /4! - • • •] n even 
(- ) (n ~ l) ' 2 [nSa(3 • a n ~ l er- l ll ! 

- n(n 2 - l 2 )5 3 a/5-o: n - 3 /3 n - 3 /3! + • • •] n odd. 
Likewise 



TV 2n a$= (-l) n /2 2n - 1 [S(al3 2n 

(2n)! 



l!(2n- 1) 



S(aP 2n ~ 2 a 2 p 2 + ■••] 



122 VECTOR CALCULUS 

7»p»-i a/3== (_l)«/2 2 «- 2 [7T(a/3) 2n - 1 

- ( 2n - ^ l TV(aB) 2n ~ 3 + • . .1 
l!(2n-2)1 1VKfxp) x J 

TV(ap) n /TVap = (-) n/2 [n5a i S-Q: n - 2 /? n - 2 /l! 

- n(n 2 - 2 2 )iS 3 a/3« n-4 /S n ~ 4 /3! + • • -J n even 
(_1)<*-*^1 - (n 2 - l^SPap-cT+p^fil + • • •] n odd. 

Since jS/a is a quaternion whose powers have the same 
axis we have (1 — 0/a) -1 = 1 + fi/a + 03/a:) 2 + • • • when 
Tfi < Ta, and taking the scalar gives the well-known 
formula 



Likewise 



S-^~= 1 + S/5/a + S(/3/a) 2 + 
a — p 



TV-^—= TVp/a + TV(p/a) 2 + 
a — p 



If we define the logarithm as in theory of functions of a 
complex variable we have 

log (1 - fi/a) = log 7(1 - fi/a) + log 17(1 - fi/a) 

= - fa - Itf/a)* - HP/a)* . 

Therefore 

log f(l - fi/a) - - Sfi/a - §S(/?/c*) 2 

Z °LZ_1 = TV log (1 - fi/a) = TVp/a + ^TV(p/a) 2 - 
a 

Again 

T{a - p)~ l = Ta' 1 - f(l - P/a)- 1 - fo^l + 

Pi(- SUp/a) TP/a + P 2 (- SUp/a) T 2 P/a + .••], 

where Pi P2 are the Legendrian polynomials. 

Evidently for coaxial quaternions we have the whole 
theory of functions of a complex variable applicable. 



VECTORS IN SPACE 123 

12. Solution of Some Simple Equations. 

(1). If ap = a then p = oT l a. 

(2) . If Sap = a then we set Vap = f where £* is any vector 
perpendicular to a, and adding, p = aa _1 + a~ l $. 

(3). If Fap = jS then *Sap = a: where # is any scalar, and 
adding we have p = a~ l (3 + aaaf" 1 . 

(4). If Vapfi = y then SaVapQ = &x 2 p/3 = <* 2 £p/3 = Say 
and SpVap(3 = /3 2 £ap = S/fy. Now 

Fap/5 = aS/3p - pSafi + (3Sap 

and substituting we have 

p = [o;- 1 ^7 + /T 1 ^ - y]/8afi. 

The solution fails if Sa(3 = 0. In this case the solution is 

p = _ a-'S^y - p^Sa-iy + xVofi, 

x any scalar. 

(5). If Yapp = 7 then Sa(3pSafi = &*07 and Soft) 
= Sa(3y/Sa(3. Adding to Va(3p, we have 

afip = 7 + Sa(3y/SaP and p = 0^or*7 + '(hcT*8cfiyl8c&. 

(6). If &xp = a, £/3p = b, then a^p = zFa/3 + V(al3 
- ba)Va(3. 

(7). If Sap = a, S(3p = b, Syp = c, then 

pSafiy = aV(3y + bVya + cFa/5. 

(8). If gag -1 = |3 then g = (x/3 + y)/(a + /3) where x and 
?/ are any scalars. Or we may write 

q = u + 0(a + |8) + wFa?/3 where u = — w#a(a: + /3). 

(9). If gag" 1 = y, q^q' 1 = 8, then 

V(y - a)(8 - ft! 



.. 



1 + 



S(T + «)(«- ft 



124 VECTOR CALCULUS 

(10). If qaq- 1 = f, qpq~ l = *, qyq~ l = f, then 

S-flft - «) - 0, S-q( V - ft - 0, flf.gtf - 7) = 0, 

hence Fg is coplanar with the parentheses, and we have 

x(i - a) + 2/(77 - ft + H(f - 7) = 
where 

»:*:*->- 2S 7 (r? - ft : 2Sy(i - a) : S(£ + a)(i, - ft. 

The six vectors are not independent. Vq is easily found 
and thence Sq from 

qa = £q. 

(11). If (p - a)" 1 + (p - ft" 1 - (P ~ 7)" 1 ~ (P ~ 5)- 1 
= 0, then if we let 

ifi' ~ aT 1 = 1 * (TO - 5)" 1 - 5] - [(a - 6)" 1 - 5]) 

= (p — 8)(p — a) _1 (« — 5), etc., 

where p', a', 0', 7' are the vectors from D, the extremity of 
5, to the inverses with respect to D, of the extremities of 
p, a, ft 7, then 

(p' - a')" 1 + (p' - ft)" 1 - (p' - 7T 1 = 0. 

Prove that 



1 - ft _ y - ft _ P ' - y _ r y - /n i/2 



p 



whence p' and p. (R. Russell.) 

(12). If (q - a)" 1 + (q - 6)" 1 - (q - c)" 1 - (q - d)~' 
= 0, we set 

(q - d)(q' - d)= (a- d){a' - d) = (b - d)(b' - d) 

= (c - d){c' - d) - 1, 



VECTORS IN SPACE 125 

thence 

(q - d)-> -(q- a)' 1 = (4 - d)-\a - d)(q - a)' 1 
(q - d)~i [(a~d)/(q-a)+(b- d)l(q- b)- (c - d)/(q- e)] 
- (?' - a')' 1 + (?' - &T 1 ~ W ~ cT 1 
and we have q' from 

(V - cW - C) = (g' - 6')/(g' - «0 

= (q' - c')l(a' - c') - [(V - c')Ka' - c')]K 

(R. Russell.) 

13. Characteristic Equation. If we write q = Sq + Vq 
and square both sides we have q 2 = S 2 q + (Vq) 2 + 2Sq-Vq 
whence 

g 2 - 2qSq + S 2 q - V 2 q = 0. 

This equation is called the characteristic equation of q. 
The coefficients 

2Sq and S 2 q - V 2 q = T 2 q 

are the invariants of q; they are the same, that is to say, 
if q is subjected to the rotation r()r -1 . They are also the 
same if Kq is substituted for q. Hence they will not define 
q but only any one of a class of quaternions which may be 
derived from each other by the group of all rotations of the 
form rQr~ l or by taking the conjugate. 
The equation has two roots in general, 

Sq + Tqyl - 1 and Sq - Tq^ - 1. 

Since these involve the V — 1 it leads us to the algebra of 
biquaternions which we do not enter here, but a few re- 
marks will be necessary to place the subject properly. 

Since the invariants do not determine q we observe that 
we must also have UVq in order to have the other two 
parameters involved. 



126 VECTOR CALCULUS 

If we look upon UVq as known then we may write the 
roots of the characteristic equation in the number field of 
quaternions as Sq + TVqUVq and Sq — TVqUVq or 

q and Kq. 

If we set q -f- r for q and expand, afterwards drop all the 
terms that arise from the identical equations of q and r 
separately, we have left the characteristic equation of two 
quaternions, which will reduce to the first form when they 
are made to be equal. This equation is 

qr+rq-2Sq-r- 2Sr-Vq + 2SqSr - 2SVqVr = 0. 

We might indeed start with this equation and develop the 
whole algebra from it. 
We may write it 

qr-\- rq- 2qSr - 2rSq + 4Sq-Sr + S-qr + S-rq = 

which involves only the scalars of q, r, qr, and rq. 

14. Biquaternions. We should notice that if the param- 
eters involved in q can be imaginary or complex then 
division is no longer unique in certain cases. Thus if 

Q 2 =q 2 
we have as possible solutions 

Q = ± q and also Q = ± V (- l)UVq-q. 

If q 2 = and Vq = then TVq = and we have 
Vq = x(i + j V — 1) where X is any scalar and i, j are any 
two perpendicular unit vectors. 



CHAPTER VII 
APPLICATIONS 

1. The Scalar of Two Vectors 
1. Notations. The scalar of the product of two vectors 
is defined independently by writers on vector algebra, as 
a product. In such cases the definition is usually given for 
the negative of the scalar since this is generally essentially 
positive. A table of current notations is given. If a and (3 
define two fields, we shall call S*cfi the virial of the two 
fields. 

S-a(3 = — a X /3 Grassman, Resal, Somoff, Peano, Bura- 
li-Forti, Marcolongo, Timerding. 

— Cfft Gibbs, Wilson, Jaumann, Jung, Fischer. 

— a/3 Heaviside, Silberstein, Foppl, Ferraris, 

Heun, Bucherer. 

— (aft) Bucherer, Gans, Lorentz, Abraham, 

Henrici. 

— a|/3 Grassman, Jahnke, Fehr, Hyde. 
Cos a/3 Macfarlane. 

[a/3] Caspary. 

For most of these authors, the scalar of two vectors, 
though called a product, is really a function of the two 
vectors which satisfies certain formal laws. While it is 
evident that any one may arbitrarily choose to call any 
function of one or more vectors their product, it does not 
seem desirable to do so. For Gibbs, however, the scalar 
is defined to be a function of the dyad of the two vectors, 
which dyad is a real product. The dyad or dyadic of 
Gibbs, as well as the vectors of most writers on vector 
analysis, are not considered to be numbers or hypernumbers. 

127 



128 VECTOR CALCULUS 

They are looked upon as geometric or physical entities, 
from which by various modes of "combination" or de- 
termination other geometric entities are found, called 
products. The essence of the Hamiltonian point of view, 
however, is the definition by means of geometric entities of 
a system of hypernumbers subject to one mode of multiplica- 
tion, which gives hypernumbers as products. Functions 
of these products are considered when useful, but are called 
functions. 

2. Planes and Spheres. It is evident that the condition 
for orthogonality will yield several useful equations, and 
of these we will consider a few. 

The plane through a point A, whose vector is a, per- 
pendicular to a line whose direction is 8 has for its equation, 
since p — a is any vector in the plane, 

S-d(p-a) = 0. 

If we set p = 8Sa/d we have the equation satisfied and as 
this vector is parallel to 5 it is the perpendicular from the 
origin to the plane. The perpendicular from a point B 
is b~ l S{a - 0)5. 

If a sphere has center D and radius T(3 where /? and — (3 
are the vectors from the center to the extremities of a 
diameter, then the equation of the sphere is given by the 
equation 

S(p - 3 + fi)(p - d - P) = 0, orp 2 - 2S8 P + 5 2 - /3 2 = 0. 

The plane through the intersection of the two spheres 

p 2 - 2£5ip + ci = = p 2 - 2S8 2P + c 2 

is 2S(5i — 5 2 )p = ci — c 2 . 

The form of this equation shows that it represents a plane 



APPLICATIONS 129 

perpendicular to the center line of the spheres. The point 
where it crosses this line is 

X18] + x 2 8 2 

P = i » 

Xi + x 2 

whence solving, we find 

p = v(h + 8 2 )-\V8,8 2 + i(cj - <*)>. 

3. Virial. If (3 is the representative of a force in direction 
and magnitude then its projection on the direction a is 
a~ 1 Sa^ f and perpendicular to this direction crWafi. If a 
is in the line of action of the force, the projection is fit If a 
is a direction not in the line of action then the projection 
gives the component of the force in the direction a. If a 
is the vector to the point of application of the force then 
Sa(3 is the virial of the force with respect to a, a term intro- 
duced by Clausius. It is the work that would be done by 
the force in moving the point of application through the 
vector distance a. If a fe an infinitesimal distance say, 
8a, then — S8a(3 is the virtual work of a small virtual dis- 
placement. The total virtual work would be 8V = 
— 2S8a n (3 n for all the forces. 

4. Circulation. In case a particle is in a vector field 
(of force, or velocity, or otherwise) and it is subjected to 
successive displacements 8p along an assigned path from 
A to B, we may form the negative scalar of the vector 
intensity of the field and the displacement. If the vector 
intensity varies from point to point the displacements 
must be infinitesimal. The sum of these products, if there 
is a finite number, or the definite integral which is the limit 
of the sum in the infinitesimal case, is of great importance. 
If a point is moving with a velocity a [cm./sec] in a field of 
force of /3 dynes, the activity of the field on the point is 



130 VECTOR CALCULUS 

— S-(3<t [ergs/sec.]. The field may move and the point 
remain stationary, in which case the activity is S-(3a. The 
activity is also called the effect, and the power. If <r is the 
vector function of p which gives the field at the point P we 
have for the sum 

- 2Sa8p or - // Sa8p. 

This integral or sum is called the circulation of the path for 
the field a. 

5. Volts, Gilberts. For a force field the circulation is the 
work done in passing from A to B. If the field is an electric 
field E, the circulation is the difference in voltage between 
A and B. If the field is a magnetic field H, then the circula- 
tion is the difference in gilbertage from A to B. It is 
measured in gilberts, the unit of magnetic field being a 
gilbert per centimeter. There is no name yet approved for 
the unit of the electrostatic field, and we must call it volt 
per centimeter. The unit of force is the dyne and of work 
the erg. 

6. Gausses and Lines. In case the field is a field of flux 
a, and the vector TJv is the outward normal of a surface 
through which the flux passes, then 

- SaUv 

is the intensity of flux normal to or through the surface 
per square centimeter. The unit of magnetostatic flux B 
is called a gauss; the unit of electrostatic flux D is called a 
line. The total flux through a finite surface is the areal 
integral 

— fSaUvdA, written also — fSadv. 

The flux-integral is called the transport or the discharge. 
Thus if D is the electric induction or displacement, the 



APPLICATIONS 131 

discharge through a surface A is — fSDUvdA, measured 
in coulombs. Similarly for the magnetic induction B, 
the discharge is measured in maxwells. 

7. Energy-Density. Activity-Density. Among other 
scalar products of importance we find the following. If 
E and D are the electric intensity in volts/cm. and induction 
in lines at a point, — |$ED is the energy-density in the 
field at the point in joules/cc. If H and B, likewise, are the 
magnetic intensity in gilberts/cm., and gausses, respectively, 
— 2^#HB is the energy in ergs. If J is the electric cur- 
rent-density in amperes/cm. 2 , — S • E J is the activity in 
watts/cc. If G is the magnetic current-density in heavi- 
sides*/cm. 2 , — S ■ H G is the activity in ergs/sec. If the 
field varies also, the electric activity is — >S- E(J + D) and 
the magnetic activity — $H(G + B). 

EXERCISES 

1. An insect has to crawl up the inside of a hemispherical bowl, the 
coefficient of friction being 1/3, how high can it get? 

2. The force of gravity may be expressed in the form a = — mgk. 
Show that the circulation from A to B is the product of the weight by 
the vertical difference of level of A and B. 

3. If the force of attraction of the earth is <r = — hUp/p 2 show 
that the work done in going from A to B is 

hiTa- 1 - T0- 1 ]. 

4. The magnetic field at a distance a from the central axis of an 
infinite straight wire carrying a current of electricity of / amperes is 

H = 0.2ia -1 (— sin di + cos 6j) (i andj perpendicular to wire) 
and the differential tangent to a circle of radius a is ( — a sin 6 i 
+ a cos 9j)dd. Show that the gilbertage is 0.2/ (0 2 — 0i) gilberts, 
which for one turn is OAirl. 

Prove that we get the same result for a square path. 

5. The permittivity k of a specimen of petroleum is 2 [abfarad/cm.], 
and on a small sphere is a charge of 0.0001 coulomb. The value of 
the displacement D at the point p is then 

D = 9^2 UplTp2 [lineg] 

* A heaviside is a magnetic current of 1 maxwell per second. 



132 VECTOR CALCULUS 

What is the discharge through an equilateral triangle whose corners 
are each 4 cm. from the origin, the plane of the triangle perpendicular 
to the field? 

6. If magnetic inductivity p. is 1760 [henry/cm.] and a magnetic 
field is given by 

H = la [gilbert/cm.], 

then the magnetic induction is 

B = 7 -1760a [gausses]. 

What is the flux through a circular loop of radius a crossing the field 
at an angle of 30°? 

7. If the velocity of a stream is given by 

<r = 24(cos 6 i -f sin dj), 

what is the discharge per second through a portion of the plane whose 
equation is Sip = — 12 from 

d = 10° to 6 = 20°? 

8. The electric induction due to a charge at the origin of e coulombs is 

D = - eUp/T P Hir [lines]. 

What is the total flux of induction through a parallelepiped whose 
center is the origin? 

9. The magnetic induction due to a magnetic point of m maxwells is 

B = - mUp/Tp 2 [gausses]. 

What is the total flux of induction through a sphere whose center is 
the point? 

10. In problem 8, if the permittivity is 2 = k, then the electric 
intensity 

E = rH>4r. 

What is the amount of energy enclosed in a sphere of radius 3 cm. and 
center at a distance from the origin of 10 cm.? 

11. In problem 9, if the inductivity is 1760 and the magnetic in- 
tensity is 

H = p~% 

how much energy is enclosed in a box 2 cm. each way, whose center 
is 10 cm. from the point and one face perpendicular to the line joining 
the point and the center? 

12. If the current in a wire 1 mm. in diameter is 10 amperes and 
the drop in voltage is 0.001 per cm., what is the activity? 



APPLICATIONS 133 

13. If there is a leakage of 10 heavisides through a magnetic area of 
4 cm. 2 , and the magnetic field is 5 gilberts/cm., what is the activity? 

14. Through a circular spot in the bottom of a tank which is kept 
level full of water there is a leakage of 100 cc. per second, the spot 
having an area of 20 cm. 2 . If the only force acting is gravity what is 
the activity? 

15. If an electric wave front from the sun has in its plane surface 
an electric intensity of 10 volts per cm., and a magnetic intensity of 
0033 gilberts per cm., and if for the free ether or for air y. = 1 and 
k = £-10~ 20 , what is the energy per cc. at the wave front? (The 
average energy is half this maximum energy and is according to Langley 
4.3 -10 -5 ergs per cc. per sec.) 

16. If a charge of e coulombs is at a point A and a magnetic point 
at B has m maxwells, what is the energy per cc. at P, any point in space, 
the medium being air? 

8. Geometric Loci in Scalar Equations. 
(1). The equation of the sphere may be written in each 
of the forms 

a/p = Kp[a, 

S(p - a)/(p + a) = 0, 

S2a/(p + a) = 1, 

S2p/(p + <*) - 1, 

T(Sp/a + Vp/a) = 1, 

Tip - ca) m T(cp - a), 
S{p - a) (a - »08 - 7)(Y - B)(S - p) - 0, 

a 2 Sfiyp + j3 2 Syap + y 2 Sa(3p = p 2 Sa(3y 
(p-aO 2 (p-/3) 2 (p-7) 2 (P-5) 2 

(p - a) 2 (a - /3) 2 (a - y) 2 (a - 5) 2 

(p-/?) 2 (/? -«) 2 (/5-t) 2 (0-S) 2 

(P-T) 2 (Y-«) 2 (Y-0) 2 (7-5) 

(p - 5) 2 (5 - a) 2 (5 - /3) 2 (5 - 7) 2 



Interpret each form. 

(2). The equation of the ellipsoid may be written in the 
forms 

S 2 p/a - V 2 p/(3 = 1, 

where a is not parallel to ft 

T(p/y + Kpjb) = T(p/8 + tfp/7), 

rOup + pX)=x 2 -/* 2 . 



134 VECTOR CALCULUS 

The planes 

a p 

cut the ellipsoid in circular sections on Tp = Tfi. These 
are the cyclic planes. Tfi is the mean semi-axis, Ufi the 
axis of the cylinder of revolution circumscribing the ellip- 
soid, a is normal to the plane of the ellipse of contact of 
the cylinder and the ellipsoid. 
In the second form let 

r 1 - -£, 7- 1 = - £> t 2 = n 2 - TJ, 

then the semi-axes are 

a=rX+7>, 6= ^~ TfX * > c=T\-T». 
T(\ - n) 

(3). The hyperboloid of two sheets is S 2 p/a + F 2 p//3 = 1. 
(4). The hyperboloid of one sheet is S 2 p/a + V 2 p/(3 = — 1. 
(5) . The elliptic paraboloid of revolution is 

SplP+V 2 p/(3 = 0. 

(6). The elliptic paraboloid is Sp/a + V 2 p/(3 = 0. 
(7). The hyperbolic paraboloid is Sp/a Sp/fi = Sp/y. 
(8). The torus is 

T(± bUarWap - p) = a, 

2bTVap = ± (Tp 2 +b 2 - a 2 ), 

4b 2 S 2 ap = 4b 2 T 2 p - (T 2 p + b 2 - a 2 ) 2 , 

Aa 2 T 2 p - 4b 2 S 2 ap = (T 2 p - b 2 + a 2 ) 2 , 

SU(p - «V (a 2 - b 2 ))l(p + cW (a 2 - b 2 )) = ± b/a, 

p = ± bJJoTWar + at/Y, r any vector. 

(9). Any surface is given by 

p = <p(u, v). 



APPLICATIONS 135 

A developable is given by p = <p(t) + ucp'it). 
(10). A cone is f(U[p - a]) = 0. 
The quadric cone is SapSfip — p 2 = 0. 
The cone through a, (3, y, 8, e is 

S-V(Va(3V8e)V(V(3yVep)V(Vy8Vpa) = 0, 

which is Pascal's theorem on conies. 

The cones of revolution through X, n, v are 

The cones of revolution which touch S\p = 0, Sfxp = 0, 
Svp = 0, are 

The cone tangent to (p — a) 2 + c 2 = from /? is 

c 2 (p -a-$) 2 = V 2 (3(p - a). 

The polar plane of /3 is £/3(p — a) — — c 2 . 
The cone tangent to 

a p 

from 7 is 

(*i- F, S- 1 )( fl, J'- p i-0 

-( S^S^-- SV?V?-- lY=0. 
\ a a a a / 

The cylinder with elements parallel to y is 

( s *i- f 1- i )H- p ?) 

_(s>sl-sv>vl) 2 = o. 

\ a a a a) 



136 VECTOR CALCULUS 

For further examples consult Joly : Manual of Quater- 
nions. 

2. The Vector of Two Vectors 
Notations, If a and /3 are two fields, we shall call V-a(3 
the torque of the two fields. 

Va(3 = Va(3 Hamilton, Tait, Joly, Heaviside, Foppl, 
Ferraris, Carvallo. 
cqS Grassman, Jahnke, Fehr. 

aX Gibbs, Wilson, Fischer, Jaumann, Jung. 
[a, /3] Lorentz, Gans, Bucherer, Abraham, Timer- 
ding. 
[a | /?] Caspary . 

a A j3 Burali-Forti, Marcolongo, Jung. 
aj8 Heun. 

Sin a/3 Macfarlane. 

Iaccb Peano. 
1. Lines. The condition that two lines be parallel is that 
Vafi = 0. Therefore the equation of the line through the 
origin in the direction a is Vap — 0. 

The line through parallel to a is Va(p — fi) = or 
Vap = Va(3 = y. The perpendicular from 5 on the line 
Vap = 7 is 

— a~ l Vab + a~ l y. 

The line of intersection of the planes, S\p = a, S^p = b, is 
VpV\fx = a/x — 6X. If we have lines Vpa — y and Vp& = 8 
then a vector from a point on the first to a point on the 
second is 5/3" 1 — 7a -1 + #/3 — ya. If now the lines in- 
tersect then we can choose x and y so that this vector will 
vanish, corresponding to the two coincident points, and 
thus 

S{bp~ l - ya~ l )$a = = S8a + Syp. 



APPLICATIONS 137 

If we resolve the vector joining the two points parallel and 
perpendicular to Vaft we have* 

5/3 -1 — ya~ l + xfi — ya 

= • (Va^S • VaP(bpr l - yoT 1 + zp - ya) 

= -(VaP)-\S5(x+ Spy) 



L a Fa/3 Fa/3 J 

L Va0 P Vap] 

- «-* f- SaPS ^~ + a 2 S JL 1 
Vap Vap] 



Hence the vector perpendicular from the first line to the 
second is 

- (Vafl-KStct + Spy) 

and vectors to the intersections of this perpendicular with 
the first and second lines are respectively 



and 



ya x — a 1 \ 8 ' — ^— 

L Va(3 J 



* Note that 

(Va0)- l V(Vu0)(z0 - ya) = xp - ya 
(y« j S)- 1 F-7a/3(5 J 3- 1 - ya~ l ) = (Vc0)- l (- a'^Sfiya - (r l S<*&) 



Va ,(-^S^ + p-S^) 



10 



138 VECTOR CALCULUS 

The projections of the vectors a, y on any three rectangular 
axes give the Pluecker coordinates of the line. For applica- 
tions to linear complexes, etc., see Joly: Manual, p. 40, 
Guiot: Le Calcul Vectoriel et ses applications. 

2. Congruence. The differential equation of a curve or 
set of curves forming a congruence whose tangents have 
given directions cr, that is, the vector lines of a vector field <r, 
is given by 

Vdpa = 

or its equivalent equation 

dp = adt. 

3. Moment. The moment of the force /3 with respect 
to a point whose vector from an origin on the line of @ is 
a, is — Fa/3. If the point is the origin and the vector to 
some point in the line of application of the force is a, then 
the moment with respect to the origin is Vafi. If the 
point is on the line of application the moment obviously 
vanishes. If several forces have a common plane then 
the moments as to a point in the plane will have a common 
unit vector, the normal to the plane. If several forces are 
normal to the same plane, their points of application in 
the plane given by ft, ft, ft, • • • , their values being a\a> 
a 2 a, a s a, • • • , then the moments are 

F(aift + a 2 ft + a 3 ft + • • •)« [dyne cm.]. 
If we set 

«ift + 02ft + 03ft + • • • = ftai + a 2 + a z + • • •)/ 

then /3 is the vector to the mean point of application, which, 
in case the forces are the attractions of the earth upon a 
set of weighted points, is called the center of gravity. If 
ai + #2 + a 3 + • • • = 0, we cannot make this substitution. 



APPLICATIONS 139 

4. Couple. A couple consists of two forces of equal 
magnitude, opposite directions and different lines of action. 
In such case the mean point becomes illusory and the sum 
of the moments for any point from which vectors to points 
on the lines of action of the forces are a h a 2 respectively, is 

V{a x - a 2 )P. 

But a\ — a 2 is a vector from one line of action to the other, 
and this sum of the moments is called the moment of the 
couple. It is evidently unchanged if the tensor of /? is 
increased and that of a\ — a 2 decreased in the same ratio, 
or vice versa. 

5. Moment of Momentum. If the velocity of a moving 
mass m is a cm./sec, then the momentum of the mass is 
defined to be ma gr. cm./sec. The vector to the mass 
being p, the moment of momentum of the mass is defined 
to be 

Vpma = mVpa [gm. cm. 2 /sec.]. 

6. Electric Intensity. If a medium is moving in a mag- 
netic field of density B gausses, with a velocity a cm./sec, 
then there will be set up in the medium an electromotive 
intensity E of value 

E=Fo-B-10~ 8 [volts/centimeter]. 

For any path the volts will be 

- fSd P E= + fSdpBa-10- 8 . 

If this be integrated around any complete circuit we shall 
arrive at the difference in electromotive force at the ends 
of the circuit. 

7. Magnetic Intensity. If a magnetic medium is moving 
in an induction field of D lines, with a velocity a, then there 
will be produced in the medium at every point a magnetic 



140 VECTOR CALCULUS 

intensity field 

H = OAwVDa [gilberts/cm.]. 

For any path the gilbertage will be OAirf SdpaD. 

8. Moving Electric Field. If an electric field of induc- 
tion, of value D lines, is moving with a velocity a, then 
there will be produced in the medium at the point a mag- 
netic field of intensity H gilberts/cm. where 

H m OAirVaD. 

For a moving electron with charge e, this will be 
— (eUp/4:irTp 2 ). For a continuous stream of electrons 
along a path we would have 

the point being the origin. 

9. Moving Magnetic Field. If a magnetic field of in- 
duction of value B gausses is moving with a velocity cr, 
it will produce at any given point in space an electric 
intensity E = V - BolO -8 volts per centimeter. 

10. Torque. If a particle of length dp is in a field of 
intensity <r which tends to turn the particle along the lines 
of force, then the torque produced by the field upon the 
element is 

V-dpa. 

If a line runs from A to B, the total torque is 

// V-dpe. 
For instance if dp, or in case of a non-uniform distribution 
cdp, is the strength in magnetic units, maxwells, of a wire 
magnet from A to B, in a field a, then 

fIV-dpa or f/V-cdpa 

is the torque of the field upon the magnet. 






APPLICATIONS 141 

11. Poynting Vector. An electric intensity E volts/cm. 
and magnetic intensity H gilberts/cm. at a point in space 
are accompanied by a flux of energy per cm. 2 R, given by 
the formula 

4xR = — — [ergs/cm. 2 sec.]. 

This is the Poynting vector. 

12. Force Density. The force density in dynes/cc. of 
a field of electric induction on a magnetic current is given by, 

F = 4ttFDG : 10 [dynes/cc], 

where D is the density in lines of electric displacement 
G is the magnetic current density in heavisides per cm. 2 . 
If the negative of F is considered we have the force per cc. 
required to hold a magnetic current in an electrostatic 
field of density D. 

The force density in dynes/cc. of a field of magnetic 
induction on a conductor carrying an electric current is 

F-ijr.il. 

A single moving charge e with velocity a will give 
F =AweVaiJiVaD. 

13. Momentum of Field. The field momentum at a 
point where the electric induction is D lines and magnetic 
induction B gausses is T = 3-10 9 V- DB [gm. cm./sec.]. If 
the magnetic induction is due to a moving electric field then 
T = 0.047rF- D/jlVDct, and if the electric induction is due 
to a moving magnetic field, 

T = VB/cVaB. 

47T-3-10 10 



142 vector calculus 

3. The Scalar of Three Vectors 

1. Area and Pressure. If we consider two differential 
vectors from the point P, say dip, d 2 p, then the vector area 
of the parallelogram they form is Vdipd 2 p. If then we have 
a distribution of an areal character, such as pressure per 
square centimeter, /3, the pressure normal to the differential 
area will be in magnitude 

— S(3dipd 2 p. 

The vector Vdipd 2 p may be represented by dp or JJvdA. 
The vector pressure normal to the surface will be 

UpS(3dipd 2 p. 

There will also be a tangential pressure or shear, which is 
the other component of /3. 

2. Flux. If j8 is any vector distribution the expression 
— S($d\pd 2 p is often called the flux of /? through the area 
Vd\pd 2 p. It is to be noted however that the dimensions 
of the result in physical units must be carefully considered. 
Thus the flux of magnetic intensity is of dimensions that 
do not correspond to any magnetic quantity. 

3. Flow. If /3 is the velocity of a fluid in cm./sec, then 
the volume passing through the differential area per second 
is 

— Sfid\pd 2 p [cc./sec.]. 

4. Energy Flux. The dimensions of the Poynting energy 
flow R show that it is the current of energy per second across 
a cm. 2 , hence the total flow per second through an area is 

-SRd^p-- 8 -™™^ [ergs/sec] 

In the case of a straight conductor carrying a current of 
electricity, we have at a distance a from the wire in a 



APPLICATIONS 143 

direction at right angles to the wire directly away from it 
the value 

T R= (4ir)- 1 10 8 JS;(0.2Ja- 1 ). 

Consequently if we consider one centimeter of wire in 
length and the circumference of the circle of radius a we 
shall have a flux of energy for the centimeter equal to 

J(ft-«0 [jouks]. 

This is the usual J 2 R of a wire and is represented by heat. 

5. Activity. For a moving conductor we have already 
expressed the vector E, and as the current density J can 
be computed from the intensity of the field (J = k E) we 
have then for the expression of the activity in watts per 
cubic centimeter of conductor 

A= -SaBhO- 8 = -S(V(rB)k(VaB)-lO- ie [watts]. 

Likewise in the case of the magnetomotive force due to 
motion and the magnetic current G = IH we have for the 
activity per cubic centimeter of circuit 

A=- SDaQ = - S-(VDa)l(VD(r)-10- 7 [watts]. 

6. Volts. The total electromotive difference between 
two points in a conductor is the line-integral along the 
conductor 

- fSdpaBlCr* [volts]. 

7. Gilberts. The total magnetomotive difference be- 
tween two points along a certain path is the line-integral 

— AirfSdpDo- [gilberts]. 

4. Vector of Three Vectors 
1. Stress. We find with no difficulty the equations 

V-a(Ua± Uy)y = ± TyTa(Ua± Uy) 
and 

V-a(Vay)y = — Say -V -ay. 



144 VECTOR CALCULUS 

If now we have a state of stress in a medium, given by its 
three principal stresses in the form 

0i = g — 7V dynes/cm. 2 normal to the plane orthogonal 

to U(U\+ Un), 
92 = g — S\fx dynes/cm. 2 normal to the plane orthogonal 

to UV\fi, 
gz = g + T\p dynes/cm. 2 normal to the plane orthogonal 
to U{U\ - Un), 

gi < gi < gz, 

then the stress across the plane normal to /? is 

V\fo + 0. 

If the scalars g it g%, g z are dielectric constants in three 
directions (trirectangular) properly chosen, then the dis- 
placement is 

D = FXE/x + gE. 

If the scalars are magnetic permeability constants, 

B = V\Hfi + g W. 
If the scalars are coefficients of dilatation, then becomes 

(T-- VWp+gp. 

If the scalars are elasticity constants of the ether, then 
according to Fresnel's theory, the force on the ether is, 
for the ether displacement ft . . 

V\fo + gp. 

If the scalars are thermoelectric constants in a crystal, 
then 

D = FXQm + gQ. where Q is the flow of heat. 

If g = the scalars are TV, - TV, - SXfi. If V\fi = 0, 
the scalars are 7V> — TV> T\p, that is, practically — t 
along X and + t in all directions perpendicular to X. 






CHAPTER VIII 

DIFFERENTIALS AND INTEGRALS 
1. DlFFEKENTIATION AS TO A SCALAR PARAMETER 

1. Differential of p. If the vector p depends upon the 
scalar parameter t, say 

p = <p(t), 

then for two values of t which are supposed to be in the 
range of possible values for t 

Pi — Pi = <p(h) — <p(ti) t 
ti — t\ t% — ^1 

If now we suppose that U < h < t 2 and that h and t 2 can 
independently approach the limit, t , then we shall call 
the limit of the fraction above, if there be such a limit, the 
right-hand derivative of p as to t, at t , and if t 2 < h < t , 
we shall call the limit the left-hand derivative of p as to 
t at t . In case these both exist and are equal, and if p 
has a value for t which is the limit of the two values of 
<p(ti), then we shall say that p is a continuous function of 
t at t and has a derivative as to t at to. 

There is no essential difference analytically between the 
function <p and the ordinary functions of a single real 
variable, and we will assume the ordinary theory as known. 

It is evident that for different values of t we may con- 
sider the locus of P which will be a continuous curve. 
Since p 2 — pi is a chord of the curve the limit above will give 
a vector along the tangent of the curve. Further the tensor 
of the derivative, Tp' = T(p'{t) y is the derivative of the 
length of the arc as to the parameter t. If the arc s is the 
parameter then the vector p' is a unit vector. 

145 



146 VECTOR CALCULUS 

EXAMPLES 

(1) The circle 

p = a cos -f sin 0, To: = Fft Sa0 = 0, 

p' = — a sin + cos 0. 

(2) The helix 

p = a cos + sin + 70, 
p' = — a sin + cos 0+7. 

(3) The conic 

_ a* 2 + 20< + 7 
P at 2 +2U+C ' 

Multiplying out, t 2 (a - ap) + 2*(0 - bp) + (7 - cp) = 
for all values of £. For 2 = 0, p = 7/c, and for t = 00 , 
p = a/a, hence the curve goes through a/a and 7/c. 
We have 

rfp/d< = [t 2 (ba — a0) + <(ca — 07) + (c0 — by)] times scalar. 

Hence for t = 0, the direction of the tangent is 0/6 — 7/c 
at 7/c, for t = 00, the direction of the tangent is 
0/6 — a/a at a/a. Since these vectors both run from the 
points of tangency to the point 0/6, the curve is a conic, 
tangent to the lines through 0/6 and the two points a/a 
and 7/c, at these two points. If the origin is taken at 
0/6, so that p = w + 0/6, and if a! = a/a — 0/6, 7' = 7/c 
- 0/6, then 

at\a! - tt) - 26/tt + c(y f - w) = 

is the equation of the curve. 

If now we let w run along the diagonal of the parallelo- 
gram whose two sides are a'y' so that tt = x(a! + y'), then 
substituting we have 

at 2 x + 2btx - c(l - x) = 0, 

at 2 (l - x) - 2btx - ex = 0. 



DIFFERENTIALS 147 

From these equations we have 

t 2 = c/a 
and 

x = Vac/2(Vac± b). 

These values of x give us the two points in which the 
diagonal in question cuts the curve. The middle point 
between these two is 

Referred to the original origin this gives for the center 

, ,, c a - 2b(3 + ay 

k= r + p b = — — =£— • 

2(ac — b 2 ) 

If we calculate the point on the curve for 

bh + e 



ah+ b 



we shall find that for the points p 2 , pi we have J(p 2 + Pi) 
= k, so that k is the center of the curve and diametrically 
opposite points have parameters 

h and t 2 = — — r-x > 
ati ~t o 

an involutory substitution. If ac = b 2 , k becomes co ex- 
cept when also the numerator = 0. [Joly, Manual, Chap. 
VII, art. 48.] 

In general the equation of the tangent of any curve is 

IT = p + Xp'. 

We may also find the derivatives of functions of p, when 
p = (p(t), by substituting the value of p in the expression 
and differentiating as before. Thus 

let p = a cos 6 + P sin 6 where Ta # Tp. 



148 VECTOR CALCULUS 

Then 

Tp = V [- a 2 cos 2 6 - 2Sap sin 6 cos 6 - 2 sin 2 6], 

We may then find the stationary values of Tp in the manner 
usual for any function. Thus differentiating after squaring 

a 2 sin 26 - 2Sa(3 cos 26 - fi 2 sin 26 = 0, 
tan 26 - 2Sap/(a 2 - /3 2 ). 

2. Frenet-Serret Formulae. Since the arc is essentially 
the natural parameter of a curve we will suppose now that 
p is expressed in terms of s, and accents will mean only 
differentiation as to s. Then both 

p and p + dsp' 

are points upon the curve. 

The derivative of the latter gives p' + dsp", which is also 
a unit vector since the parameter is s. Thus the change in 
a unit vector along the tangent is dsp", and since this 
vector is a chord of a unit circle its limiting direction is 
perpendicular to p', and its quotient by ds has a length whose 
limit is the rate of change of the angle in the osculating plane 
of the tangent and a fixed direction in that plane which 
turns with the plane. That is to say, p" in direction is 
along the principal normal of the curve on the concave side, 
and in magnitude is the curmture of the curve, which we 
shall indicate by the notation 

Unit tangent is a = p', 

Unit normal is |9 = Up", curvature is Ci = Tp", 

Unit binormal is y = Va(3, so that Ciy = Vp'p". 

The rate of angular turn of the osculating plane per centi- 
meter of arc is found by differentiating the unit normal of 
the plane. Thus we have 

Ti = cf 2 hW - Fp'p"-c 2 ].' 



DIFFERENTIALS 149 

But d 2 = T 2 p" = - Sp"p" and therefore Cl c 2 = - S P "p f ". 
Substituting for c 2 we have 

71 - cr 3 [- Sp"p"Vp'p f " + SpV'Wl 
= cr z [Vp , Vp"Vp ,,, p"] 

= crWaVc 1 (3Vp'"c 1 p 

= crW-aPVp"^ = cr l VyVp'"p = cr l pSyp"' 

= - «lft 

where «i is written for the negative tensor of 71 and is the 
tortuosity. It may also be written in the form 

Again since /? = ya we have at once the relations 

j3i = 7i« + 7«i = «i7 ~~ C\a. 

Thus we have proved Frenet's formulae for any curve 

«i = erf, ft = ai7 — ci«, 71 = — a$. 

It is obvious now that we may express derivatives of any 
order in terms of a, ft y, and Oi, Ci, and the derivatives of 
ai and Ci. 

For instance we have 

Pi = OL, p 2 = fci, 

Pa = ftci + Pc 2 = fe + (701 — aci)ci, 
Pi = 0c 3 + 2{yai — aci)c 2 + (ya 2 — ac 2 )ci 

- ^( ai 2 + Cl 2 ) Cl . 

The vector w = aai + 7C1 is useful, for if 77 represents in 
turn each one of the vectors a, /3, 7, then 771 = Fa^ It is 
the vector along the rectifying line through the point. 
The centre of absolute curvature k is given by 

K = p - lip" m p + Pld. 



150 VECTOR CALCULUS 

The centre of spherical curvature is given by 

a = k + yd/da • c{~ 1 = k — yc 2 /aiCi 2 . 

The polar line is the line through K in the direction of 7. 
It is the ultimate intersection of the normal planes. 

3. Developable s. If we desire to study certain de- 
velopables belonging to the curve, a developable being the 
locus of intersections of a succession of planes, we proceed 
thus. The equation of a plane being S(w — p)rj = 0, 
where t is the vector to a variable point of the plane, and 
p is a point on the curve, while rj is any vector belonging 
to the curve, then the consecutive plane is 

S(t - p)f) + ds'd/dsS(w - p)r) = 0. 

The intersection of this and the preceding plane is the line 
whose equation is 

7r = p + (— r)Sar) + t)lVr}r}i. 

This line lies wholly upon the developable. If we find a 
secOnd consecutive plane the intersection of all three is a 
point upon the cuspidal edge of the developable, which is 
also the locus of tangents of the cuspidal edge. This vector 
is 
tv = p + (VwySar} + 2Vr)7]iSar)i + Vr}7}iS^rjCi)/ST]r}ir]2' 

By substituting respectively for 77, a, ft 7, we arrive at the 
polar developable, the rectifying developable, the tangent- 
line developable. 

EXAMPLE 
Perform the substitutions mentioned. 

4. Trajectories. If a curve be looked upon as the path 
of a moving point, that is, as a trajectory, then the param- 
eter becomes the time. In this case we find that (if 
p = dp/dt, etc.) the velocity is p = av, the acceleration is 






DIFFERENTIALS 151 



p = ficiv 2 + av. The first term is the acceleration normal 
to the curve, the centrifugal force, the second term is the 
tangential acceleration. In case a particle is forced to 
describe a curve, the pressure upon the curve is given by 
(3civ 2 . There will be a second acceleration, p = a(v — wi 2 ) 
+ (3(2cii + c 2 v) + yaiCiV. The last term represents a 
tendency per gram to draw the particle out of the osculating 
plane, that is, to rotate the plane of the orbit. 

5. Expansion for p. If we take a point on the curve 
as origin, we may express p in the form 

p = sa + %cis 2 (3 — %s*(ci 2 a — c 2 /3 — cmy) 

— ^ 4 (3c 2 cia: ~~ £I C3 ~~ c * — Clttl2 l ~~ T[2c 2 ai + da 2 ]) 

EXERCISES 

1. Every curve whose two curvatures are always in a constant ratio 
is a cylindrical helix. 

2. The straight line is the only real curve of zero curvature every- 
where. 

3. If the principal normals of a curve are everywhere parallel to a 
fixed plane it is a cylindrical helix. 

4. The curve for which 

Ci = 1/ms, ai = 1/ns, 
is a helix on a circular cone, which cuts the elements of the cone under 
a constant angle. 

5. The principal normal to a curve is normal to the locus of the 
centers of curvature at points where Ci is a maximum or minimum. 

6. Show that if a curve lies upon a sphere, then 

cr 1 = A cos a + B sin a = C cos (a + e), A, B, C, e are constants. 
The converse is also true. 

7. The binormals of a curve do not generate the tangent surface of 
a curve. 

8. Find the conditions that the unit vectors of the moving trihedral 
afiy of a given curve remain at fixed angles to the unit vectors of the 
moving trihedral of another given curve. 

Two Parameters 
6. Surfaces. If the variable vector p depends upon two 
arbitrary parameters it will terminate upon a surface of 



152 VECTOR CALCULUS 

some kind. For instance if p = <p(u, v), then we may 
write for the total differential of p 

dp = dud/du(<p) -f- dvd/dv((p) = du<p u + dv<p v . 

We find then 

Fdp = £dw 2 + 2Fdudv + GW, 
where 

E = — ^ tt 2 , F = — S<p u <p v , G = — ^t, 2 . 

We have thus two differentials of p, one for » = constant, 
one for u = constant, which will be tangent to the para- 
metric curves upon the surface given by these equations, 
and may be designated by 

pidu, p 2 dv. 

The normal becomes then 

v = v Pl p 2 , Tv = V (EG - F 2 ) = H. 

For certain points or lines v may become indeterminate, 
the points or lines being then singular points or singular 
lines. 

7. Curvatures. If we consider the point p and the 
point p + dupi -f- dvp 2 the two normals will be 

v and v + duV(p n p2 + P1P12) 

-f- dvV(pi2p2 + P1P22) + • • • 
which may be written 

v and v + dv. 

The equations of these lines are 

V(w - P )v =0, V(w- p- d P )(v + dv) = 0. 

They intersect if 

Sdpvdv = 0. 

Points for which this equation holds lie upon a line of 



DIFFERENTIALS 153 

curvature so that this is the differential equation of such 
lines. If we expand the total differentials we have 

du 2 Spivi>i + 2dudvS(piw 2 + Pivv\) + dv 2 Sp 2 w 2 = 0. 

We may also write the equation in the form 

dp + xv + ydv = = pidu -\- p 2 dv + xv + yv\du + yv 2 dv. 

Multiply by (pi + yv\){p 2 + yv 2 ) and take the scalar part 
of the product, giving 

S(pi + yvi)(pi + P2#> = o 

= y 2 Svviv 2 + 2ySv{piv 2 + ^ip 2 ) + ^ 2 . 

The ultimate intersection of the two normals is given by 

t = p + dp + yv + y<&>, 

that is by yv. Hence we solve for yTv, giving two values 
R and R f which are the principal radii of curvature at the 
point. The product and the sum of the roots are re- 
spectively 

RR' = yy'Tv 2 - Tv%- Sw 1 v 2 ), 
R + R' = — 2TvSv(piv 2 -\- vip 2 )/Swiv 2 . 

The reciprocal of the first, and one-half the second divided 
by the first, that is, 

— Spvivt/v 4 and Sv(piv 2 + vip 2 )/Tv*, 

are the absolute curvature and the mean curvature of the 
surface at the point. 

The equation of the lines of curvature may be also written 

vSdpvdv = = V-VdpVvdv = VdpV(dv/vv) = VdpdUv. 

Hence the direction of dUv is that of a line of curvature, 
when du and dv are chosen so that dp follows the line of 
curvature. That is, along a line of curvature the change 
li 



154 VECTOR CALCULUS 

in the direction of the unit normal is parallel to the line 
of curvature. 

When the mean curvature vanishes the surface is a 
minimal surface, the kind of surface that a soapfilm will 
take when it extends from one curve to another and the 
pressures on the two sides are equal. The pressure indeed 
is the product of the surface tension and twice the mean 
curvature, so that if the resultant pressure is zero, the 
mean curvature must vanish. If the radii are equal, as in 
a sphere, then the resultant pressure will be twice the 
surface tension divided by the radius, for each surface of the 
film, giving difference of pressure and air pressure = 4 
times surface tension/radius. The difference of pressure 
is thus for a sphere of 4 cm. radius equal to the surface 
tension, that is, 27.45 dynes per cm. 

When a surface is developable the absolute curvature is 
zero, and conversely. Surfaces are said to have positive 
or negative curvature according as the absolute curvature 
is positive or negative. 

EXERCISES 

1. The differential equation of spheres is 

Vp(p - a) = 0. 

2. The differential equations of cylinders and cones are respectively 

Sva = 0, Sv(p - a) = 0. 

3. The differential equation of a surface of revolution is 

Sapv = 0. 

4. Why is the center of spherical curvature of a spherical curve not 
of necessity the center of the sphere? 

5. Show how to find the vector to an umbilicus (the radii of curvature 
are equal at an umbilicus). 

6. The differential equation of surfaces generated by lines that are 
perpendicular to the fixed line a is 

SVav<pVocv = 0, 

where <p is a linear function. 



DIFFERENTIALS 155 

7. The differential equation of surfaces generated by lines that meet 
the fixed line V(p — (3) a = is 

SVvV{p - P)a<p{V V V(p - 0)a) = 0. 

8. The differential equation of surfaces generated by equal and 
similarly situated ellipses is 

SV(Va&-p)v(VYa0-p) = 0. 

9. Show that the catenoid 

p = xi + a cosh x/a(cos 8j + sin 6k) 

is a minimal surface, and that the two radii are db Tv, the normal which 
is drawn from the point to the axis. 

2. Differentiation as to a Vector 
1. Definition. Let q = /(p) be a function of p, either 
scalar, vector, or quaternion. Let p be changed to p + dt • a 
where a is a unit vector, then the change in q is given by 

dq= q' - q = f{p+ dt-a) - /(p), 
and 

dq/dt = Lim [/(p + dta) - f(p)]/dt 

as dt decreases. If we consider only the terms in first 
order of the infinitesimal scalar dt we can write 

dq = dtf(p, a) 

in which a will enter only linearly. 

In a linear function of a however we can introduce the 
multiplier into every term in a and write dta = dp, so that 
we have dq a linear function of dp, 

dq = f'(p, dp). 

It needs to be noted that the vector a is a function of the variable dt, 
although a unit vector. The differential of q is of course a function of 
the direction of dp in general, but the direction may be arbitrary, or be 
a function of the variable vector p. It may very well happen that the 
limit obtained above may be different for a given function / according 
to the direction of the vector a. In general, we intend to consider the 



156 VECTOR CALCULUS 

vector dp as having a purely arbitrary direction unless the contrary is 
stated. 

EXAMPLES 

(1) Let 

q = " P 2 . 
Then 

dq = - [p2 + 2dtS-pa - p 2 ] = - 2dtSpa = - 2Spdp. 

Also since q = T 2 p we have 

dq = 2TpdTp=- 2Spdp, 
whence 

dTp/Tp = Sdp/p, or dTp = - SUpdp. 

(2) From the definition we have 

d(qr) = dq-r + g-dr, 
hence 

d(Tp-Up) = dTp-Up+ Tp-dUp = dp 

and utilizing the result of the preceding example, we have 

dUp/Up = Vdplp. 

Also we may write dUp = — Vdpp-p/T 3 p = pVdpp/T 3 p 
= p~ l VpdpjTp, etc. This equation asserts that the dif- 
' ferential of Up is the part of the arbitrary differential of p 
perpendicular to Up, divided by the length of p, that is, 
it is the differential angle of the two directions of p laid off 
in the direction perpendicular to p in the plane of p and 
dp. In case dp is along the direction of p itself, 

dUp = 0. 

(3) We have since 

d{pp~ l p) = dp = dp-p~ l p + pd{p~ l )p + pp~ l dp 

= 2dp + pd(p' l )p 9 



DIFFERENTIALS 157 

and thence 

dp = — pd{p~ l )p, 

i.p-i = - p-Hpp- 1 = [p-'Spdp - p-WpdpWFp 

= p-'dp-p/rp. 

That is, the differential of p~ l is the image of dp in p divided 
by the square of Tp. 
Hence 

diVap)- 1 = (Vap)- l Vadp-VapjTWap. 

This vanishes if dp is parallel to a. 

(4) If x = — a 2 /p then dir = — a 2 p~ l dppj T 2 p, and for two 
different values of dp, as dip, dip, we have 

diir/diTT = p~ l dipld\p-p. 

Therefore in the process of "inverting" or taking the 
"electrical image " we find that the biradial of two dif- 
ferential vectors is merely reflected in p. Interpret this. 

(5) T- = c is a family of spheres with a and — a as 

p — a 

limit points. For a differential dp confined to the surface 
of any sphere we have then 

Sdp[(p + a)- 1 - (p - a)- 1 ] = 0. 

A plane section through a can be written Syap = 0, in 
which Syadp = gives a differential confined to the plane. 
Therefore a differential tangent to the line of intersection 
of any plane and any sphere will satisfy the equation 

Vdp[VVya«p + a)- 1 - (p - a)" 1 )] - 0. 

But the expression in the () is a tangent line to any sphere 
which passes through A and — A. For the equation of 
such a sphere would be 

p 2 - 2Sadp - a 2 = 



158 VECTOR CALCULUS 

where 5 is any vector, hence for any dp along the sphere, 
S(p — VaS)dp = 0. But (p + a) -1 — (p — a) -1 is parallel 
to a(p 2 + « 2 ) - 2pSap and 5(p - Va8)[a(p 2 + a 2 ) 
— 2piSap] = — Sap[p 2 — a 2 — 2£pa5]. For points on the 
sphere the [] vanishes, hence the vector in question is a 
tangent line. Also Vttt is perpendicular to it or r, therefore 
the differential equation above shows that the tangent dp 
of the intersection of the plane and the sphere of the 
system is perpendicular to a sphere through A and —A. 
Hence all spheres of the set cut orthogonally any sphere 
through A and —A. 

(6) The equation SU = e is a familv of tores pro- 
p—a 

duced by the rotation of a system of circles about their 
radical axis. From this we have 

SU(p + a)(p-a) = -e, 

VU(p + a)(p - a) - V (1 - e 2 )UVap = a. 

Differentiating the scalar equation we have 

L P+ OL 

+ TJ(p + a)V-^- >U(p - a)l = 

P — OL J 

or 

Sadp[(p + a)- 1 - (p - a)- 1 ] = 0. 

Now in a meridian section a is constant so that 

Vdp[(p + a)" 1 - (p - a)" 1 ] - 

and dp is for such section tangent to a sphere through 
A and —A. 

EXERCISES 
1. The potential due to a mass m at the distance Tp is m/Tp in 



DIFFERENTIALS 159 

gravitation units. Find the differential of the potential in any direc- 
tion, and determine in what directions it is zero. 

2. The magnetic force at the point P due to an infinite straight 
wire carrying a current a is H = — 2h/Vap. Find the differential of 
this and determine in what direction, if any, it is zero. For Vdpa = 0, 
dH = 0; for dp = dsVa^Vap, dH = - Hds/TV<r P ; for dp = dsUVap, 
dH - V<rUd8./TV<rp. 

3. The potential of a small magnet a at the origin on a particle of 
free magnetism at p is u = Sap/T 3 p. Find the variation in directions 

Up, UVap, UaVap. 

4. The attraction of gravitation at a point P per unit mass in gravita- 
tion units is 

a = - Up/T*p. 

Find the differential of <x in the directions Up and F/3p. 

da = - {pHp - SpSpdp)/T 5 P ; parallel to p, - 2/p 3 ; 

perpendicular, UV@p/T s p. 

5. The force exerted upon a particle of magnetism at p by an element 
of current a at the origin is 

H = - V<x P IT s p. 

Then dH = {pWadp - 3Va P Spdp)/T 5 P ; in the direction of p, 37a/p 3 ; 
in the direction Vap, — VaUVap/T 3 p. 

6. The vector force exerted by an infinitesimal plane current at 
the origin perpendicular to a, upon a magnetic particle or pole at p is 

a = (ap 2 - SpSap)/T* P . 

Find its variation in various directions. 

2. Differential of Quaternion. We may define differen- 
tials of functions of quaternions in the same manner as 
functions of vectors. Thus we have T 2 q — qKq so that 

2TqdTq = d(qKq) = [(q + dtUq)(Kq + dtUKq) - qKq] 
= dtlqUKq + UqKq] 
= qKdq -f- dqKq 
= ZSqKdq = 2SdqKq. 
That is, 

dTq = SdqUKq = SdqUq' 1 = TqSdq/q 
or 

dTq/Tq = Sdq/q. 



160 VECTOR CALCULUS 

In the same manner we prove the other following formulae. 

dUq/Uq = Vdq/q, dSq = Sdq, dVq = Vdq, 

dKq = Kdq, S(dUq)/Uq = 0, 

dSUq = SUqV(dq/q) = - S(dq/qUVq')TVUq 

= TVUqdzq, 
dVUq = VUKqV(dqlq), 

dTVUq = - SdUqUVq = SUqdzq, 

d-q 2 = 2Sqdq + 2Sq- Vdq + 2Sdq- Vq, 

d-q~ l = — q~ l dqq~ l , 

d-qaq- 1 = - 2V -qdq^qVaq- 1 = 2V-dq(Va)q-\ 

that is, if r = gag -1 , then 

dr = 2V(dqjq>r) = - 2V(q-dq- l -r) 

= 2V(Vdq/q)r - 2q-V 'V{q- l dq-a)q~ l 
dUVq= V'Vdq/Vq-UVq, 
dzq= S[dqKUVq-q)]. 

We define when 7a = 1 

a x = cos • irx/2 + sin • 7nc/2 • a = catf • %tx; 
thus 

d-a* = tt/2-o:^ 1 ^. 

If Ta # 1, then 

d-a x = dz[log 7W* + tt/2 -a x+1 /Ta\, 

3. Extremals. For a stationary value of /(p) in the 
vicinity of a point p we have ay(p) = 0. If /(p) is to be 
stationary and at the same time the terminal point of p 
is to remain on some surface, or in general if p is to be subject 

*Tait, Quaternions, 3d ed., p. 97. 



DIFFERENTIALS 161 

to certain conditioning equations, we must also have, if 
there is one equation, q(p) = 0, dq(p) = 0, and if there are 
two equations, g(p) = and h(p) = 0, then also dg(p) = 0, 
dh(p) = 0. Whether in all these different cases /(p) attains 
a maximum of numerical value or a minimum, or otherwise, 
we will consider later. 

EXERCISES 

1. g( p ) = (p — a) 2 -f- « 2 = 0, find stationary values of Tp = f(j>). 
Differentiating both expressions, 

Sdp(p —a) = = Sdpp, 

for all values of dp. Hence we must have dp parallel to V • tp where t is 
arbitrary, and hence Srp(p — a) = 0, for all values of r. Therefore 
we must have Vp(p — a) = 0, or Yap = 0, or p = ya. Substituting 
and solving for y, 

y = 1 ± a/Tcc, p = a ± aUa. . 

2. g( p ) = ( p — «) 2 + a 2 = 0. Find stationary values of &/3p. 
Sdp(p - a) = = *S/3ap, whence dpP.WjS, £'T,3(p - a) = 0, 

7/3(p - a) = 0. 

p - a =y0, y = a/T(3, p = a ± at//3. 

3. ^( p ) = (p — a )2 -f a 2 = 0, &G>) = *S/3p = 0, find stationary values 
of Tp. 

Sdp( P - a) = = Spdp = Spdp, whence S-p0(p — a) « - £pa/S, 
and since £p0 = 0, p = yV-fiVafi. 

p = V0VaP(l ± V[a 2 - S*a0)/TVal3). 

4. #(p) = p 2 — SapSpp + a 2 = 0. Find stationary values of Tp. 

£d P p = o = £dp(p - «S/8p - jSflap), 

p = x(aS$p + /8/Sap) = (a£/3p + pSap)/(Sa(3 ± Ta0), 

whence 

Sap = TaSpU/3, 



= SpU(3(Ua ± U0)/(SUaU0 T 1). 



Substituting in the first equation, we find SpUp, thence p. 
5. Sfip = c, >Sap = c', find stationary values of Tp. 

Sd P p = Sadp = Spdp = 0, p = xa + y$ and 

z£a/3 -f 2//3 2 = c, xa 2 + ySafi = c', whence x and y. 



162 VECTOR CALCULUS 

6. Find stationary values of Sap when (p — a) 2 -f- a 2 = 0. 

Sctdp = = Sdp(p - a); 
hence 

p = ya = a ± aJ7a 
and 

Sap = a 2 ± aTa. 

7. Find stationary values for Sap when p 2 — SppSyp + a 2 = 0. 

Sadp = = £d P (p - )857P - ySfip), 
P =xa+ fiSyp + ySfip, etc. 

8. Find stationary values of TV8p when 

(p - a) 2 + a 2 = 0. 

9. Find stationary values of SaUp when 

(p - a) 2 + a 2 = 0. 

10. Find stationary values of SaUpSpUp when 

Syp + c = 0. 

4. Nabla. The rate of variation in a given direction of 
a function of p is found by taking dp in the given direction. 
Since df(p) is linear in dp it may always be written in the 
form 

where $ is a linear quaternion, vector, or scalar function 
of dp. In case / is a scalar function, $ takes the form 

— Sdpv, 

where v is a function of p, which is usually independent of 
dp. In case v is independent of the direction of dp, we 
call / a continuous, generally differentiable, function. 
Functions may be easily constructed for which v varies 
with the direction of dp. If when v is independent of dp we 
take differentials in three directions which are not in the 
same plane, we have 

pS - dipd 2 pd 3 p = V'dipd 2 p-Sd 3 pp + V '• d 2 pd 3 p • Sdipp 

+ V - d 3 pdipSd 2 pp 
= — V 'd 1 pd 2 p'd 3 f '— Vd 2 pd 3 p-dif 

— V-d 3 pdip-d 2 f. 



DIFFERENTIALS 163 

It is evident that if we divide through by Sdipdipdzp, the 
different terms will be differential coefficients. The entire 
expression may be looked upon as a differential operation 
upon/, which we will designate by V. Thus we have 

v= V/ = 

_ ( Vdipdip - d z + V- d 2 pd s p • di + V- d^pdip ■ d 2 ) ,, , 
S • dipdipdzp 
We may then write 

df( P ) = - SdpVfip). 

If the three differentials are in three mutually rectangular 
directions, say i, j, k, then 

V = id/dx + jd/dy + kd/dz. 

It is easy to find V/ for any scalar function which is gener- 
ally differentiate from the equation for df(p) above, that 
is, df(p) = — SdpVf. For instance, 

VSap = - a, Vp 2 = - 2p, VTp = Up, 
V(Tp) n = nTp n - l Up = nTp n ~ 2 -p, V TVap = TJVap-a, 
VSaUp = - p-WUpa, V • SapSpp = - pSd$ - Vap(3, 

V-log TVap = -^~, 
vap 

VT(p - a)-' = - U(p - a)lT\p- a), 
VSaUpS(3Up = p-WpVap^P, 
Vlog Tp= U P /Tp= -p~\ 

1 



V(ZpA*) = - p~ 1 UVpa = 



pUVap 



5. Gradient. If we consider the level surfaces of /(p), 
/(p) = C, then we have generally for dp on such surface or 
tangent to it S dp p = = df(p) where p. is the normal of the 



164 VECTOR CALCULUS 

surface. Since Sdp\7f — and since the two expressions 
hold for all values of dp in a plane 

M = *V/, 

or since the tensor of p. is arbitrary, we may say V/(p) is 
the normal to the level surface of /(p) at p. It is called 
the gradient of /(p), and by many authors, particularly in 
books on electricity and magnetism, is written grad. p. 

The gradient is sometimes defined to be only the tensor 
of V/, and sometimes is taken as — V/. Care must be 
exercised to ascertain the usage of each author. 

Since the rate of change of /(p) in the direction a is 
— &*V/(p), it follows that the rate is a maximum for the 
direction that coincides with UVf, hence the gradient 

V/(p) 

gives the maximum rate of change off(p) in direction and size. 
That is, TVf is the maximum rate of change of /(p) and 
UVfis the direction in which the point P must be moved in 
order that /(p) shall have its maximum rate of change. 

6. Nabla Products. The operator V is sometimes called 
the Hamiltonian and it may be applied to vectors as well 
as to scalars, yielding very important expressions. These 
we shall have occasion to study at length farther on. It 
will be sufficient here to notice the effect of applying V and 
its combinations to various expressions. It is to be ob- 
served that VQ may be found from dq, by writing dq 
= $-dp, then VQ = i$i + j$j + k$k. 

For examples we have 

Vp = {Vdipdzp-dzp + Vd 2 pdsp-dip 

+ Vdzpdip • d 2 p) I '(— Sdipd 2 pd 3 p) 

= - 3 

since the vector part of the expression vanishes. 



DIFFERENTIALS 165 

Vp _1 = — (Vdipd 2 p-p~ 1 d 3 pp~ 1 + •••)/(— Sd 1 pd 2 pd 3 p) 
Since 



- - P" 2 . 



dUp = V^ • Up, dTp = - SUpdp. 



Hence 



VUp = 2iV--Up= -~, VTp= Up. 
p Tp 

Expressions of the form 2F(i, i, Q) are often written 
F($ > r> Q)> a notation due to McAulay. 

Vap = a, 

VfaSfap + cx 2 S/3 2 p + mSfop) = - 0m + /5 2 a 2 + 1830:3), 

VFap = 2a, VVap(3 = &xft 

VSapVfip = - Sapp + 3/3£a<p - pSa(3, 

VVaUp= (a + p^Sap)/ Tp, V • TTap = C/Fap • a, 

VTVpVap - (Fap + ap)UVpVap, 

VVap/T 3 p m (ap 2 - SpSap)/T 5 p, 

V ' UV « P =Tkp> VUVpVap = ^P-, 

V(Vap)-i=0, V (g)=0. 

EXERCISE 

Show that (Fa/3 -<l>y -+- y0y<£>a + Vy<x'3?P)/Sa0y is independent 
of a, /3, 7, where $ is any rational linear function (scalar, vector, or 
quaternion) of the vector following it. If <*> = S8( ) + 2ai<S/3i( ) the ex- 
pression is 5 + S/Siai. 

Notation for Derivatives of Vectors 
Directional derivative 
- SaV, Tait, Joly. 
a- V, Gibbs, Wilson, Jaumann, Jung. 

Tp -a, Burali-Forti, Marcolongo. 



166 VECTOR CALCULUS 

Circuital derivative 
VaV, Tait, Joly. 
a X V, Gibbs, Wilson, Jaumann, Jung. 

Projection of directional derivative on the direction. 
S-<r l vSau, Tait, Joly. 

— > Fischer. 
da 

Projection of directional derivative perpendicular to the 

direction 

V-trhi'SV'a, Tait, Joly. 

—— * Fischer. 
da 

Gradient of a scalar 

V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo, 

Bucherer. 
grad, Lorentz, Gans, Abraham, Burali-Forti, Marcolongo, 

Peano, Jaumann, Jung. . 

— grad, Jahnke, Fehr. 

[Fischer's multiplication follows Gibbs, d/dr 
d p. , being after the operand, the whole being 

dr read from right to left; e.g., Fischer's 

Vfl is equiv. to — vSV.] 

Gradient of a vector 

V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo. 
grad, Jaumann, Jung. 

-=- > Fischer. 
dr 

7. Directional Derivative. One of the most important 
operators in which V occurs is— SaV, which gives, the 



DIFFERENTIALS 167 

rate of variation of a function in the direction of the unit 
vector a. The operation is called directional differentiating. 

SaV'Sfo = - SaP, SaV-p 2 = - 2Sap, 
SaVTp - SaUp, SaVTp- 1 = - Sap/Tp* = UY^p- 2 , 

SaVTVap= 0, SaV-Up= - ^~ • 

An iteration of this operator upon Tp~ l gives the series of 
rational spherical and solid harmonics as follows : 

- SaVTp- 1 = - Sap/Tp* = UYiTff*, 
Sl3VSaVTp- 1 = (3SapS(3p+ Tp 2 Sa(3)Tp- 5 = 2\Y 2 Tp~\ 
SyVSWSaVTp- 1 = - (3.5SapS(3pSyp 

+ 32S(3ySapTp 2 )Tp- 7 = 3\Y 3 Tp~\ 

For an n axial harmonic we apply n operators, giving 

Y n = S.(- l) 8 (2n - 2s)!/[2 n -*nl(n - s)l\ES n - 28 aUpS s a 1 a 2 , 

^ s^ n/2. 

The summation runs over n — 2s factors of the type 
SaiUpSoi2Up • - • and s factors of the type SajCtjSotnar - - , 
each subscript occurring but once in a given term. The 
expressions Y are the surface harmonics, and the expressions 
arising from the differentiation are the solid harmonics 
of negative order. When multiplied by Tp 2n+1 we have 
corresponding solid harmonics of positive order. 

The use of harmonics will be considered later. 

8. Circuital Derivative. Another important operator is 
Va\7 called the circuital derivative. It gives the areal 
density of the circulation, that is to say, if we integrate 
the function combined with dp in any linear way, around 
an infinitesimal loop, the limit of the ratio of this to the 
area of the loop is the circuital derivative, a being the normal 
to the area. We give a few of its formulae. We may also 



168 VECTOR CALCULUS 

find it from the differential, for if dQ = $dp, Fa V • Q 

VaV • Tp - VaUp, FaV • Tp n = nTp n ~ 2 Vap t 

VaV - Up = (Sap 2 - pSap)/Tp\ VaV-SQp = F/3a, 

Fa V • V(3p = a(3+ S-aP, FaV -ft> = 2Sa(3, 

FaV • 7Tft> - - V-apUVpp, FaV -p - - 2a, 

Fa V • (aiSftp + a 2 »S/3 2 p + a 3 S/3 3 p) - Sa(« A + "A 

+ a 3 fo) + FaiFa/3i + Fa 2 Fa/3 2 + Fa 3 Fa/3 3 . 

9. Solutions of VQ = 0, V 2 Q =0. In a preceding 
formula we saw that V(Vap)~ l = 0. We can easily find 
a number of such vectors, for if we apply Sa V to any vector 
of this kind we shall arrive at a new vector of the same 
kind. The two operators V and Sa V • are commutative 
in their operation. For instance we have 

d(Vap)~ l = - (VapyWadp-iVap)- 1 ; 
hence 

T = ^V-(Fap)- 1 = {Vap)- l V$a>{Vap)- 1 

is a new vector which gives Vr = 0. The series can easily 
be extended indefinitely. Another series is the one de- 
rived from Up/T 2 p. This vector is equal to p/T 3 p, and its 
differential is 

(-p 2 dp+SSdpp.p)/T% 

The new vector for which the gradient vanishes is then 

(-ap 2 +3Sap-p)/7V 

The latter case however is easily seen to arise from the 
vector V Tp~ l , and hence is the first step in the process of 
using V twice, and it is evident that S7 2 Tp~ l = 0. So also 
the first case above is the first step in applying V 2 to log 
TVap-a~ l so that V 2 (log TVap-a) = 0. Functions of p 
that satisfy this partial differential equation are called 



DIFFERENTIALS 169 

harmonic functions. That is,/(p) is harmonic if V 2 /(p) = 0. 
Indeed if we start with any harmonic scalar function of p 
and apply V we shall have a vector whose gradient van- 
ishes, and it will be the beginning of a series of such vectors 
produced by applying &*iV, Sa 2 V, • • • to it. However we 
may also apply the same operators to the original harmonic 
function deriving a series of harmonics. From these can 
be produced a series of vectors of the type in question. 
V 2 • F(p) is called the concentration of F(p) . The concentra- 
tion vanishes for a harmonic function. 

EXERCISES 

Show that the following are harmonic functions of p: 
1. Tp- 1 tan" 1 Sap/Spp, 

where a and /? are perpendicular unit vectors, 



2. 


Tf* log tan ^ Z £ 


3. 
where 

and 


tan -1 Sap/S/3p 

Sa(3 = 
a 2 = £2 = _ 1# 


4. 


logtan^ z - • 

£j CL 



10. Harmonics. We may note that if u, v are two scalar 
functions of p, then 

V -uv = u Vfl + v\7u 
and thus 

V 2 -uv = u\7h + vV 2 u + 2SVuVv. 

Hence the product of two harmonics is not necessarily 
harmonic, unless the gradient of each is perpendicular to 
the gradient of the other. 
Also if u is harmonic, then 

\7 2 -uv = u\7 2 v + 2SVu\7v. 

12 



170 VECTOR CALCULUS 

If u is harmonic and of degree n homogeneously in p, then 
w/7p 2n+I is a harmonic of degree — (n + 1). For 

V 2 (fp 2n+1)-1 . V[ _ ( 2n+ l) p r p -2n-3] 

= - (2n+ l)(2n)Tp~ 2n - 3 
and 
SVuVTp- 2 "- 1 = - (2n+ l)Tp- 2n -*SVup 

= (2n+ l)(2n)uTp- 2n -*; 
hence 

V 2 -u/Tp 2n+1 - 0. 

In this case w is a solid harmonic of degree n and uTp~ 2n ~ l 
is a solid harmonic of degree — n — 1. Also uTp" 11 is a 
corresponding surface harmonic. The converse is true. 
EXAMPLES OF HARMONICS 

Degree n = 0; <p = tan -1 — - 

£>pp 

where Sc& = 0, a 2 = /3 2 = - 1; 

^ = log cot ^/ -a 2 = - 1; 
a 

S-a(3UpSapS(3p/V 2 -a(3p; 

Sa(3UpS(a + 0)pS(a - /3)p/F 2 a/3p. 

The gradients of these as well as the result of any opera- 
tion Sy V are solid harmonics of degree — 1, hence multiply- 
ing the results by Tp[n = 1, 2n — 1 = 1] gives harmonics 
again of degree 0. These will be, of course, rational 
harmonics but not integral. 

Taking the gradient again or operating by $71 V any 
number of times will give harmonics of higher negative 
degree. Multiplying any one of degree — n by Tp 2n ~ 1 
will give a solid harmonic of degree n — 1. 

Degree n = — 1. Any harmonic of degree divided by 
Tp, for example, 

1/Tp, ip/Tp, f/Tp, Saf3UpSaUpS(3p/V 2 a(3p, • • • , 



DIFFERENTIALS 171 

Degree n = — 2. 

SaUp/p 2 , <pSa(3Up/p 2 , xPSa(3Up/p 2 + P" 2 • • • . 
Degree n = 1. 

Sop, *>&*ft>, ^Softa + 7p • • • . 

Other degrees may easily be found. 

11. Rational Integral Harmonics. The most interesting 
harmonics from the point of view of application are the 
rational integral harmonics. For a given degree n there 
are 2n + 1 independent rational integral harmonics. If 
these are divided by Tp n we have the spherical harmonics 
of order n. When these are set equal to a constant the 
level surfaces will be cones and the intersections of these 
with a unit sphere give the lines of level of the spherical 
harmonics of the given order. A list of these follow for 
certain orders. Drawings are found in Maxwell's Electricity 
and Magnetism. 

Rational integral harmonics, Degree 1. Sap, S(3p, Syp, 
a, ft, y a trirectangular unit system. 

Degree 2. SapS(3p, SfoSyp, SypSap, 3S 2 ap + p 2 , S 2 ap 

- s 2 p P . 

These correspond to the operators 7p 5 [£ 2 7V, SyVSaV, 
SyVSPV, S(a + 0) VS(a - 0) V, SaVSQV] on Tp'K 

Degree 3. Representing Sap by — x, Sfip by — y, Syp by 
— z, SaV by — D x , S(3V by — D y , SyV by — D z we have 

2z 3 — 3x 2 z — 3y 2 z, 4:Z 2 x — x* — y 2 x, A.z 2 y — x 2 y — y 3 , 

x 2 z — y 2 z, xyz, x z — 3xy 2 , 3x 2 y — y 3 

corresponding to 

7)3 7)3 7)3 7) 3 _ 7) 3 7) 3 _ Q7) 3 

^ zzz ) -lszzx , Lf zzy , ^xxz > J^xyz , ^xxx > OU X yy , 

7) 3 _ Q7) 3 

■LSyyy j OJ^xxy • 



172 VECTOR CALCULUS 

Degree 4. 

3z 4 + 3y 4 + 8z 4 + 6*y - 24z 2 z 2 - 24yV, 

*z(4z 2 - Sx 2 - 3y 2 ), yz(4z 2 - 3^ - 3i/ 2 ) 

(^ _ y 2 )(6z 2 — x 2 — y 2 ), xy(6z 2 — x 2 — y 2 ), 

xz(x> - Sy 2 ), yzQx 2 - y 2 ), x* + y* - My 2 , 

xyix 2 - y 2 ) 





7) 4 7) 4 7) 4 
is zzzz ) *-* zzzx y -L/zzzy 


D 4 - 

■LS ZZXX 


. T) 4 7) 4 7) 4 _ OT) 4 

*s zzyy j M* zzxy i J^xxxz *->±s X yyz j 


7) 4 

1J yyyz 


_ Q7) 4 7) 417) 4_ ft/) 4 
oiyxxyz ) Uxxxx T ^ yyyy ^^xxyy y 




D 4 — 7) 4 

J^xxxy ^xyyy • 



The curves of the intersections of these cones with the 
unit sphere are inside of zero-lines as follows : 

Degree 1. Equator, standard meridian, longitude 90°. 

Degree 2. Latitudes ± sin -1 JV 3, equator and standard 
meridian, equator and longitude 90°, longitude ± 45°. 
Standard meridian and 90°. 

Degree 3. Latitudes 0°, db sin -1 V 0.6, latitudes ± sin -1 
V 0.2 and standard meridian, latitudes ± sin -1 V 0.2 and 90° 
longitude, equator, longitude ± 45°, equator, longitudes 
0°, 90°. Longitudes ± 30°, 90°, longitudes ± 60°, 0°. 

12. Variable System of Trirectangular Unit Vectors. 
We will consider next a field which contains at every point 
a system of three lines which are mutually perpendicular. 
That is, the lines in one direction are given by a, say, at the 
same point another set by ft and the third set by y. Each 
is a given function of p, subject to the conditions 

a/3 = 7, /?7 = a, ya = /?, a 2 = (3 2 = y 2 = — 1. 

For example, in the ordinary congruence, a being the unit 
tangent at any point of one line of the congruence, then 
the normal and the binormal are determined and would 
be ]S and 7. However /? and 7 may be other perpendicular 



DIFFERENTIALS 173 

lines in the plane normal to a. If we follow the vector 
line for /3 after we leave the point we shall get a determinate 
curve, provided we consider a to be its normal. We may 
however draw any surface through the point which has 
a for its normal and then on the surface draw any curve 
through the point. All such curves can serve as ft curves 
but a might not be their principal normal. It can happen 
therefore that the j8 curves and the y curves may start out 
from the point on different surfaces. However a, (3, and y 
are definite functions of the position of the point P, with 
the condition that they are unit vectors and mutually 
perpendicular. 

If we go to a new position infinitesimally close, a becomes 
a + da, ft becomes fi + dp, and y becomes y + dy. The 
new vectors are unit vectors and mutually perpendicular, 
hence we have at once 

S-ada = S-pdp = S>ydy = 0, Sadp = - S(3da, n . 
Spdy = - Sydp, Syda = - Sady. {L) 

These equations are used frequently in making reductions. 
We have likewise since a 2 = — 1, 

Va-a = - VW, V/3-/3 = - VW, (2) 

vy-t = - v'rr'j 

where the accent on the V indicates that it operates only 
on the accented symbols following. Similarly we have 

Va-j8 + V(3-a= - V'a0' - V'j&x', etc. (3) 

We notice also that 

S-a(SQV)a = 0, 
S-a(SQV)0 = - S-p(S()V)a, etc. (4) 

We now operate on the equation y = afi with V, and 



174 VECTOR CALCULUS 

remember that for any two vectors X/x we have X/x = — juX 
+ 2<SX/x, whence 

V7 = Va-j3 + V'aP' = Va-/3 - V/3-a + 2V'Sa(3'. (5) 

The corresponding equations for the other two vectors are 
found by changing the letters cyclically. 
Multiply every term into y and we have 

Vt-7 = Vo-a + Vj8-|8 + 2V'Sct(3'-y. (G) 

If now we take the scalar of both sides we have 

SyVy = SaVa + Sj8V0 + 2SyV'Sa(3'. (7) 

We set now 

2p = + &*Va + SjSVjS + #7 Vt (8) 

and the equation (7) gives, with the similar equations 
deduced by cyclic interchange of the letters, 

SyVSctP - - SyV'Sa'Q = - p + S7V7, 
SaV'SPy' m - SaV'Sfi'y = - P + 5a V«, 
SpVSya' = - SpV'Sy'a = - p + S0V/3, 
- S-Tf- 5a V • y] = 5a V • £77' = |&* V ■ 7 2 = 0, ( j 
- 5-a[- 5aV-7] = - SaV-Sa'y 

= Sy(— SaV -a) = Sy(u(3 + vy) = — v. 

That is to say, the rate of change of y, if the point is moved 
along a, is ]8(5aVa — p). Likewise 

dfi/ds = — 7(— p + 5aVa)— -ya. 

The trihedral therefore rotates about a with the rate 
(p — SaVa) as its vertex moves along a. Now we let 

t a = + p - SaVa. (10) 

We may also write at once, similarly, 

h - + V - S0VA * 7 = + p - 5 7 V7, (10) 
from which we derive 

t + V+ <»-+■* (ID 



DIFFERENTIALS 175 

It is also evident that 

*. + U = SrVy, t fi + *, - SaVa, / 7 + / a = 5/3 V/3. (12) 

The expressions on the left hold good for any two per- 
pendicular unit vectors in the plane normal to the vector 
on the right, and hence if we divide each by 2 and call the 
result the mean rotatory deviation for the trajectories of the 
vector on the right, we have 

TjSctVct = mean rotatory deviation for a. 

Again the negative rotation for the trajectory gives 
what we have called previously the rotatory deviation of a 
along j3. Hence, as a similar statement holds for y, the 
mean rotatory deviation is one half the sum of the rotatory 
deviations. Hence %Sa\7a is the negative rate of rotation 
of the section of a tube of infinitesimal size, whose central 
trajectory is a, about a, as the point moves along a. Or 
we may go back to (9) and see that 

SaVa = (+ p ~ SPVB) + (+ V ~ SyVy) 

= - SpV'Sya' + SyV'Sfa', 

which gives the rotatory deviations directly. 
The scalar of (5) and the like equations are 



SVa = SyVP - Sj3\7y, SVP = SaVy - SyVa, 



(13) 



SVy = SfiVa - SaVP, 
We multiply next (5) by a and take the scalar, giving 



SyVa = - SaV'Sfia' = SaV'Sa(3 f , 
SfiVa m - SaVSay* = SaV'Sya', 
SaVP = - SpV'Sy? = St3V'S(3y', 
£ T V/3 = - SpV'Spa' = S(3V'Sa(3', 
SfiVy = - SyV'Say' = SyV'Sya', 
SaVy = - SyV'Sy(3' m SyV'Sfiy'. 



(14) 



176 VECTOR CALCULUS 

We can therefore write 

SVa = - SWSPa' - SyV'Sya', 

that is SVa equals the negative sum of the projection of 
the rate of change of a along (3 on /3, and the rate of change 
of a along y on y. But these are the divergent deviations 
of a and hence — SVa is the geometric divergence of the 
section. It gives the rate of the expansion of the area of 
the cross-section of the tube around a. We may write the 
corresponding equations of /8 and y. 
Again we have 

FVa = — aSaVa — (3S(3Va — ySyVoc 

= cx(t a - v) - PSy(SaV-a) + ySp(SaV-a) 
= a(t a — p) — Va(SaV-a). 

Now from the Frenet formulae 

— Sa'V 'Ol = c a v, 

where c a is the curvature of the trajectory and v is the 
principal normal. Hence 

Wa = a(t a - p) + CJh (15) 

where /i is the binormal of the trajectory. We find there- 
fore that VVd consists of the sum of two vectors of which 
one is twice the rate of rotation of the section or an elemen- 
tary cube about a, measured along a, and the other is twice 
the rate of rotation of the elementary cube about the 
binormal measured along the binormal.* But we will see 

* This should not be confused with the rotation of a rigid area mov- 
ing along a curve. The infinitesimal area changes its shape since each 
point of it has the same velocity. As a deformable area it rotates (i.e. 
the invariant line of the deformation) with half the curvature as its 
rate. The student should picture a circle as becoming an ellipse, 
which ellipse also rotates about its center. 



DIFFERENTIALS 177 

later that this sum is the vector which represents twice the 
rate of rotation of the cube and the axis as it moves along 
the trajectory of a. Hence this is what we have called 
the geometric curl. 

We may now consider any vector a defining a vector 
field not usually a unit vector. Since a = TaUa, we have 

SVa = SUaVTa + TaSvUa. 

The last term is the geometric convergence multiplied by 
the length of a, that is, it is the convergence of a section 
at the end of a. The first term is the negative rate of 
change of TV along a. The two together give therefore 
the rate of decrease of an infinitesimal volume cut off from 
the vector tube, as it moves along the tube. In the lan- 
guage of physics, this is the convergence of a. Similarly 
we have 

Wa= VvTaU<r+ TaWUa. 

The last term is the double rate of rotation of an elementary 
cube at the end of a, while the first term is a rotation about 
that part of the gradient of Ta which is perpendicular to 
Ua. It is, indeed, for a small elementary cube a shear of 
one of the faces perpendicular to Ua, which gives, as we 
have seen, twice the rate of rotation corresponding. Con- 
sequently VVa is twice the vector rotation of the elemen- 
tary cube. 

EXAMPLES 

(1) Show that 

aSVa + (3S V0 + yS Vt 

= - VaWa - V(3WP - VyV\7y. 

(2) Show that if dipt) is zero VaWa = 0. This is the 
condition that the lines of the congruence be straight. It 
is necessary and sufficient. 



178 VECTOR CALCULUS 

(3) Let Wot - f, - SaVa « z, then Tf = V [c 2 + *% 

fi = — &x V • £ = #ia + c^/3 + c%y t where the subscript 
1 means differentiation as to s, that is, along a line of the 
congruence. 

- S^ - cip; a! = cr'Sfei + x, 
or 

This gives the torsion in terms of the curl of a and its 
derivative. 

(4) If the curves of the congruence are normals to a set 
of surfaces, then 

a = UVu and V« = V 2 u/TVu - V(l/TVu)-Vu. 

Hence we have at once SaVa = = x. This condition 
is necessary and sufficient. 

(5) If also VaWoi — 0, we have a Kummer normal 
system of straight rays. In this case by adding the two 
conditions, aV\/a = 0, that is, Wot = 0. This condition 
is also necessary and sufficient. 

(6) If the curves are plane, «i = or Sa\7a = $/3V/3 
+ SyVy or $/?£i = — xci, which is necessary and suffi- 
cient. 

(7) If further they are normal to a set of surfaces S8VP 
+ SyVy = = jS|8f i. The converse holds. 

(8) If Ci is constant, Sy£i = and conversely. 

If also plane, and therefore circles, #/3£i = or £i = X\a 
+ C\x(3. This is necessary and sufficient. 
For a normal system of circles we have also 

VVa = const = C\y. 

(9) For twisted curves of constant curvature £i = — ciaifi. 



differentials 179 

Notations 
Vortex of a vector 
VVu, Tait, Joly, Heaviside, Foppl, Ferraris. 

V X u, Gibbs, Wilson, Jaumann, Jung. 

curl u, Maxwell, Jahnke, Fehr, Gibbs, Wilson, Heaviside, 

Foppl, Ferraris. Quirl also appears. 
[Vm], Bucherer. 

rot u, Jaumann, Jung, Lorentz, Abraham, Gans, Bucherer. 
J rot u, Burali-Forti, Marcolongo. 

— ; — , Fischer. 
dr 

Vort u, Voigt. 

(Notations corresponding to VVu are also in use by 

some that use curl or rot.) 

Divergence of a vector 

— SVu, Tait, Joly. S\7u is the "convergence" of Max- 

well. 

V • u, Gibbs, Wilson, Jaumann, Jung. 

div u, Jahnke, Fehr, Gibbs, Wilson, Jaumann, Jung, 
Lorentz, Bucherer, Gans, Abraham, Heaviside, Foppl, 
Ferraris, Burali-Forti, Marcolongo. 

\7u, Lorentz, Abraham, Gans, Bucherer. 

— ~ , Fischer. 
dr 

Derivative dyad of a vector 

- SQV-u, Tait, Joly. 

• Vw, Gibbs, Wilson. 

• V ; u, Jaumann, Jung. 

du 

-p= t Burali-Forti, Marcolongo. 

aJr 

— , Fischer. 
dr 

D u - , Shaw. 



180 VECTOR CALCULUS 

Conjugate derivative dyad of a vector 

— VS«(), Tait, Joly. 
Vm-, Gibbs, Wilson. 
V; u- f Jaumann, Jung. 

Ki(), Burali-Forti, Marcolongo. 

-j-, Fischer. 
dr c 

D u -,Shaw. 

Planar derivative dyad of a vector 
WVuQ, Tait, Joly. 
VX(mX 0), Gibbs, Wilson. 
V *u, Jaumann, Jung. 

du 
CK , Burali-Forti, Marcolongo. 

— x(D u ), Shaw. 

Dispersion. Concentration 

— V 2 , Tait, Joly. V 2 is the "concentration" of Maxwell. 
V 2 , Lorentz, Abraham, Gans, Bucherer. 

V-V, Gibbs, Wilson, Jaumann, Jung. 
div grad, Fehr, Burali-Forti, Marcolongo. 

— div grad, Jahnke. 

A2, for scalar operands, 1^, ,*,, A . % - . 
A/, for vector operands, jBurah-Forti, Marcolongo. 

-7-5 > Fischer. 
dr 

Dyad of the gradient. Gradient of the divergence 

— VSV, Tait, Joly. 
VV-, Gibbs, Wilson. 
V; V, Jaumann, Jung. 

grad div, Buroli-Forti, Marcolongo. 



DIFFERENTIALS 181 

Planar dyad of the gradient. Vortex of the vortex 

VVVV(), Tait, Joly. 

V*V, Jaumann, Jung. 

rot 2 , Lorentz, Bucheoer, Gons, Abraham. 

curl 2 , Heaviside, Foppl, Ferraris. 

rot rot, Burali-Forti, Marcolongo. 

13. Vector Potential, Solenoidal Field. If £ = VVv, 
then we say that a is a vector potential of £. Obviously 

£v£ = SV 2 <r = 0. 

The vector potential is not unique, since to it may be added 
any vector of vanishing curl. When the convergence of a 
vector vanishes for all values of the vector in a given region 
we call the vector solenoidal. If the curl vanishes then 
the vector is lamellar. 

We have an example of lamellar fields in the vector field 
which is determined by the gradient of any scalar function, 
for WVu = 0. 

In case the field of a unit vector is solenoidal we see from 
the considerations of § 12 that the first and second divergent 
deviations of any one of its vector lines are opposite. If 
then we draw a small circuit in the normal plane of the 
vector line at P and at the end of dp a second circuit in 
the normal plane at p + dp, and if we project this second 
circuit back upon the first normal plane, then the second 
will overlie the first in such a way that if from P a radius 
vector sweeps out this circuit then for every position in 
which the radius vector must be extended to reach the 
second circuit there is a corresponding position at right 
angles to it in which it must be shortened by an equal 
amount. It follows that the limit of the ratio of the areas 
of the two circuits is unity. Hence if such a vector tube is 
followed throughout the field it will have a constant cross- 



182 VECTOR CALCULUS 

section. In the general case it is also clear that SVcr gives 
the contraction of the area of the tube. 

When <r is not a unit vector then we see likewise that 
SVcr by § 12 has a value which is the product of the con- 
traction in area by the TV -f- the contraction of TV multi- 
plied by the area of the initial circuit. Hence SVv repre- 
sents the volume contraction of the tube of a for length TV 
per unit area of cross-section. When the field is solenoidal 
it follows that if TV is decreasing the tubes are widening 
and conversely. 

For instance, S\/Up = — 2/Tp signifies that per unit 
length along p the area of a circuit which is normal to p 
is increasing in the ratio 2/Tp, that is, the flux of Up is 
increasing at the rate of 2/ Tp along p. Also £ • Vp = — 3 
indicates that an infinitesimal volume taken out of the 
field of p is increasing in the ratio 3. Of this the increase 2 
is due to the widening of the tubes, as just stated, the 
increase 1 is due to the rate at which the intensity of the 
field is increasing. If the field is a velocity field, the rate 
of increase of volume of an infinitesimal mass is 3 times 
per second. 

It is evident now if we multiply SVo" by a differential 
volume dv that we have an expression for the differential 
flux into the volume. If a is the velocity of a moving mass 
of air, say unit mass, then SV<? is the rate of compression 
of this moving mass, and SVcrdv is the compression per 
unit time of this mass, and fffSVcrdv is the increase 
in mass per unit time of matter at initial density or com- 
pression per unit time of a given finite mass which occupies 
initially the moving volume furnishing the boundary pf 
the integral. 

If r is the specific momentum or velocity of unit volume 
times the density, then SVr is the condensation rate or 



DIFFERENTIALS 183 

rate of increase of the density at a given fixed point, and 
SVrdv is the increase in mass in dv per unit time. Hence 
SffSVrdv is the increase in mass per unit time in a 
given fixed space. 
Since 

1 

a — -t 
c 

where c is density at a point, 

SVo- = --SVct + -SVt 
cr c 

e, „ i . B log c d log c 

= _ S(TV . logc+ _JL_ = _iL 

= total relative rate of change of density 

due to velocity and to time, 
= relative rate of change of density at a 
moving point. 
SVc-dv= increase in mass of a moving dv divided 
by the original density. 
fffSVv-dv = increase in mass in a moving volume per 
unit of time divided by original density, 
= decrease in volume of an original mass. 

For an incompressible fluid SVcr = or a is solenoidal, 
and for a homogeneous fluid SVt = or t is solenoidal. 
In water of differing salinity #Vcr = 0, SVr =\= 0. We 
have a case of constant r in a column of air. If we take 
a tube of cross-section 1 square meter rising from the ocean 
to the cirrus clouds, we may suppose that one ton of air 
enters at the bottom, so that one ton leaves at the top, but 
the volume at the bottom is 1000 cubic meters and at the 
top 3000 cubic meters. Hence the volume outflow at the 
top is 2000 cubic meters. In the hydrosphere a and r 



184 VECTOR CALCULUS 

are solenoidal, in the atmosphere r is solenoidal. We 
measure a in m?/sec and r in tons/ra 2 sec. At every sta- 
tionary boundary <r and r are tangential, and at a surface 
of discontinuity of mass, the normal component of the 
velocity must be the same on each side of the surface, as 
for example, in a mass of moving mercury and water. 

It is evident that if a vector is solenoidal, and if we 
know by observation or otherwise the total divergent devia- 
tions of a vector of length TV, then the sum of these will 
furnish us the negative rate of change of TV along a. 
Thus, if we can observe the outward deviations of r in the 
case of an air column, we can calculate the rate of change of 
TV vertically. If we can observe the outward deviations of 
a tube of water in the ocean we can calculate the decrease 
in forward velocity. 

EXERCISES. 

1. An infinite cylinder of 20 cm. radius of insulating material of 
permittivity 2 [farad/cm.], is uniformly charged with l/207r electrostatic 
units per cubic cm. Find the value of the intensity E inside the rod, 
and also outside, its convergence, curl, and if there is a potential for 
the field, find it. 

2. A conductor of radius 20 cm. carries one absolute unit of current 
per square centimeter of section. Find the magnetic intensity H inside 
and outside the wire and determine its convergence, curl, and potential. 

14. Curl. We now turn our attention to another meaning 
of the curl of a vector. We can write the general formula 
for the curl 

W<t= -aSUaVUa- pSyVTa + y(cT(T+ SfiVTa) 

Let Ua = a'. These terms we will interpret, one by one. 
It was shown that the first term is a multiplied by the sum 
of the rotational deviations of <r' . But if we consider a 
small rectangle of sides t)dt = dip and rdu = d 2 p, then the 
corresponding actual deviations are 

Sdipd 2 a f and — Sd 2 pdia' 



DIFFERENTIALS 185 

and the sum becomes 

Sdipdtff' — Sd 2 pdi<r'. 

But d 2 a' is the difference between the values of a' at the 
origin and the end of d 2 p, and to terms of first order is the 
difference of the average values of a' along the two sides 
dip and d\p + d 2 p — dip. Likewise dia is the difference 
between the average values of a' along the side d 2 p and its 
opposite. Hence if we consider Sdpa' for a path consisting 
of the perimeter of the rectangle, the expression above is 
the value of this Sdpa' for the entire path, that is, is the 
circulation of <j' around the rectangle. Hence the coefficient 

- SUaVUa 

is the limit of the quotient of the circulation around dip d 2 p 
divided by dtdu or the area of the rectangle. 

If we divide any finite area in the normal plane of a into 
elementary rectangles, the sum of the circulations of the 
elements will be the circulation around the boundary, and 
we thus have the integral theorem 

fSdpa = ffSdipd 2P V\7<j 

when Vdipd 2 p is parallel to Fy<r. The restriction, we shall 
see, may be removed as the theorem is always true. 
The component of V\7<r along a is then 

— Ua Lim j^Sdpcr/area of loop 

as the area decreases and the plane of the loop is normal to a. 
Consider next the term — (3SyV Ta. It is easy to reduce 
to this form the expression 

[- S*'(SyV)<r + Sy(S&'V)<r][- j8]. > id ; 
But this is the circulation about a small rectangle in the 

13 



ISC. VECTOR CALCULUS 

plane normal to /?. Hence the component of VVcr in the 
direction is 

— (3 Lim J'Sdpff/aresi of loop in plane normal to /?. 

Likewise the other term reduces to a similar form and the 
component of V\7<r in the direction 7 is 

— 7 Lim tfSdpa/sLYea of loop in plane normal to 7. 

It follows if a is any unit vector that the component of 
V\7(T along a is 

— a Lim JfSdpa/sLfesL of loop in plane normal to a 

as the loop decreases. The direction of UVS/a is then 
that direction in which the limit in question is a maximum, 
and in such case TV\7a is the value of the limit of the cir- 
culation divided by the area. That is, TVS/v is the maxi- 
mum circulation per square centimeter. 

Another interpretation of VV<? is found as follows: Let 
us suppose that we have a volume of given form and that a 
is a velocity such that each point of the volume has an inde- 
pendent velocity given by a. Then the moving volume will 
in general change its shape. The point which is originally 
at p will be found at the new point p + cr(p)dt. A point 
near p, say p + dp, will be found at p + dp + a(p + dp)dt, 
and the line originally from p to p + dp has become instead 
of dp, 

dp -f- dt[a(p + dp) — <r(p)] = dp — SdpV 'vdt. 

But this can be written 

dp' = dp- [W-^'dpa' + idpSVo- - iV(W(r)dp]dt. 

This means, however, that we can find three perpendicular 
axes in the volume in question such that the effect of the 



DIFFERENTIALS 187 

motion is to move the points of the volume parallel to these 
directions and to subject them to the effect of the term 

dp + iV(W(r)dp dt. 

Now if we consider an infinitesimal rotation about the 
vector e its effect is given by the form (du being half of the 
instantaneous angle) 

(1 + edu)p(l — edu) = p + 2Vepdu; 

hence the vector joining p and p + dp will become the vector 
joining p + 2Vepdu and p + 2Vepdu + dp -f- 2Vedpdu, 
that is, dp becomes dp + 2Vedp du. We find therefore 
that the form above means a rotation about the vector 
UVV<r of amount \TV\7adt, or in other words V\/a, 
when a is a velocity, gives in its unit part the instanta- 
neous axis of rotation of any infinitesimal volume moving 
under this law of velocity, and its tensor is twice the angu- 
lar velocity. For this reason the curl of a is often called the 
rotation. When V\/<r = 0, a has the form a = \/u, and u 
is called a velocity potential. If a is not a velocity, we 
still call u a potential for a. 

EXERCISES. 

1. If a mass of water is rotated about a vertical axis at the rate of 
two revolutions per second, find the stationary velocity. What are the 
convergence and the curl of the velocity? Is there a velocity potential? 

2. If a viscous fluid is flowing over a horizontal plane from a central 
axis in such way that the velocity, which is radial, varies as the height 
above the plane, study the velocity. 

3. Consider a part of the waterspout problem on page 50. 

15. Vortices. Since VVc is a vector it has its vector 
lines, and if we start at any given point and trace the vector 
line of FVo" such line is called a vortex line. The field of 
FVc is called a vortex field. If a vector is lamellar the 
vector and the field are sometimes called irrotational. The 



188 VECTOR CALCULUS 

equation of the vortex lines is 

VdpWa = - 8dp V a - V'Sdpa' - - da - V'Sdpa'. 

The rate of change of a then along one of its vortex lines is 
— V'Saa'. Since SvV^a — 0, the curl of a is always 
solenoidal, that is, an elementary volume taken along the 
vortex lines has no convergence but merely rotates. 

The curl of the curl is VvVVa = VV — S/SVa and 
thus if a is harmonic the curl of the curl is the negative 
gradient of the convergence, and if the vector is solenoidal, 
the curl of the curl is the concentration VV. 
EXERCISES 

1. If Sa<r = = SaV '<r, and if we set <r = V-ar, and determine X 
so that V-X" = t, then Xa is a vector potential of the vector <r. 

2. Determine the vector lines in the preceding problem for a. Also 
show that the derivative of X in any direction perpendicular to a is 
equal to the component of a perpendicular to both. What is V 2 A^? 

3. If a = wy and — Sy V • w = 0, then either Xa or F/3 will be 
vector potentials of <r where (iy = a and all are unit vectors and 
SyV'X =0 = SyVY. 

4. If the lines of <r are circles whose planes are perpendicular to y 
and centers are on p = ty, and To = f(TVyp), then any vector parallel 
to y whose tensor is F(TVyp), where — f = dF/dTVyp is a vector 
potential of a. Is a solenoidal? 

5. If the lines of <r are straight lines perpendicular to y and radiating 
from p = ty and T<r. = f(TVyp), then what is the condition that <r be 
solenoidal? If Ta = /(tan -1 TVyp/Syp) a cannot be solenoidal. 

6. If a =/(*Sap, S0p)-Vyp-y, then what is FV<r? Show that if/ 
is a function of tan -1 Sap/Spp, that SypVf is a function of the same 
angle, but if / is a function of TVyp, SypV •/ = and no vector of 
the form a = f(TVyp)Vyp-y can be a potential of yTVyp. If 
M = Sap/Sfip, then/0*) = - ./V0*)eW0* 2 + 1). 

7. What are the lines of a = f(Sap, Sfip) Vyp and what is the curl? 
If / is a function of TVyp, so is the curl, and if 

F{TVyp) = (TVyp)- 2 fTVyp<pTVypdTVyp 

then F-TVyp is a vector potential of the solenoidal vector y<pT{Vyp). 
If / is a function of p. the curl is a function of p., and \f(ji) Vyp is a vector 
potential of 7/O*). 

8. If <r is solenoidal and harmonic the curl of its curl is zero. If its 



DIFFERENTIALS 189 

lines are plane and it has the same tensor at all points in a line per- 
pendicular to the plane, then it is perpendicular to its curl. 

9. The vector <r = f- Up, where / is any scalar function of p, is not 
necessarily irrotational, but SaVv = 0. 

10. If a vector is a function of the two scalars S\p, Sup where X, p. 
are any two vectors (constant), or if S\p = 0, then what is true of 

11. If S<rV<r 4= 0, show that if F is determined from S\7<rVF 
= — SaX7 9 then F is the scalar potential of an irrotational vector r 
which added to <r gives a vector a', &cr'V V = 0. Is the equation for 
F always integrable? 

12. The following are vectors whose lines form a congruence of 
parallel rays f(p)a, f(Sap)a, f(Vap)a, [where/ is a scalar function], which 
are respectively neither solenoidal nor lamellar, lamellar, solenoidal. 
The case of both demands that To = constant. 

13. Examples of vectors of constant intensity but varying direction 
are 

o- = aUp, aVocp +«V(6 2 - a 2 V 2 ap). 

Determine whether these are solenoidal and lamellar. 

14. If the lines of a lamellar vector of constant tensor are parallel 
rays, it is solenoidal. If the lines of a solenoidal vector are parallel 
straight lines, it is lamellar. 

15. An example of vectors whose convergences and curls are equal 
at all points, and whose tensors are equal at all points of a surface, are 

a(x + 2yz) + &(y + Szx) + xyy, and 2yza + Szx/3 -f- y(xy + 2z) 
and the surface is 

x 2 + y 2 - z 2 + 6xyz = 0. 

Therefore vectors are not fully determined when their convergences and 
curls are given. What additional information is necessary to determine 
an analytic vector which does not vanish at oo .' Determine a vector 
which is everywhere solenoidal and lamellar and whose tensor is 12 
for Tp m oo . 

16. Show that 

— eV 2 <Z = lim r=0 [average value of q over a sphere of radius r, less the 

value at the center] divided by r 2 . 

— \V 2 q = average of (- SaV) 2 q in all directions a. 

— xVV 2 g = lim r =o [excess of average value of q throughout a small 

sphere over the value at the center} divided by r 2 . 

17. Show by expansion that 

a(p + 8p) = a(p) - S8pX7 -(r(p) 

- VS P [- Sa8p + ±S8pV P Sa8p] - W8pVSJ p* 
= VVS P [~ |F5pa + iSSpVpVSpa] - ±8pSV P <r. 



190 VECTOR CALCULUS 

i 

The first expansion expresses <r in the vicinity of p in terms of a gradient 
of a scalar and an infinitesimal rotation. The second expresses a in 
the form of a curl and a translation. 

18. Show that for any vector <r we have 

£V(W'V"&r\r"<r/7V) =0, 

where the accents show on what the V acts, and are removed after the 
operation of the accented nabla. The unaccented V acts on what is 
left. (Picard, Traits, Vol. I, p. 136.) 

19. If a, <r 2 are two functions of p, and d<n = <pi(dp),da 2 = widp), 
show that 

&<riV -SaiV — S<r 2 V -SaiV = S(<pi<r 2 — <p 2 <Ti)^7 . 

16. Exact Differentials. If the expression Sadp is the 
differential of a function u(p), then it is necessary that 
Sadp = — SdpVu, for every value of dp, which gives 

a = — Vw. 

When a is the gradient of a scalar function of u(p), u is 
sometimes called a force-function. It is evident at once 
that 

VS7<r = 0, or £FOV)cr = for every v. 

This is obviously a necessary condition that Sadp be an 
exact differential, that is, be the differential of the same 
expression, u, for every dp. It is also sufficient, for if 
VVa = 0, it will, be shown below that a = Vu, and 
SVudp = — du. 

In general if Q(p) is a linear rational function of p, 
scalar or vector or quaternion, then to be exact, Q(dp) must 
take the form 

Q(dp) — — SdpV -R(p) for every dp. 
Hence formally we must have the identity 

C()= -S()V-R(p). 
But if we fill the ( ) with the vector form VvV , we have 
Q(Vi>S7) = for every v. 



DIFFERENTIALS 191 

This may be written in the form 

Q'VV'l ) .- identically. 

EXERCISES 

1. Vadp is exact only when a = a a constant vector. For VaV\7 v = 
for every v, that is S\(vSS7p- — VSav) = for every X, v, and for X 
perpendicular to v therefore SXS/Sav = 0, or Sdav = for every v 
perpendicular to the dp that produces da. Again if X = v, 

SV* + SvVSav = 0, 

for every v. Therefore S\/a = and Sv\7 Sav = 0, or Sdav = for 
every dp in the direction of v. Hence da = for every dp and a = a 
a constant. 

2. Examine the expressions 

S^, V(Vap)dp, F.&. 

Integrating Factor 
If an expression becomes ezactf &?/ multiplication by a 
scalar function of p, let the multiplier be m. Then 

mQ(W) = 0, 
where V operates on m and Q, or 

QWm() + mQVV() = 0, 

where V operates on m only in the first term and on Q 
only in the second. This gives for Sadp 

SaVmi ) + mS( )Vo- = 0, or VaVm + mVV<r = 0. 

This condition is equivalent however to the condition 

Sa\7<7 = 0. 

Conversely, when this condition holds, we must have 

VVa- = V(tt, 

where r is arbitrary, hence StVv = 0, and Sa\7r = 0. 
But r is any variable vector conditioned only by being 



192 VECTOR CALCULUS 

perpendicular to FV<r, hence we must have for all such 
VVt — 0, or a = 0. The latter is obviously out of the 
question and hence VVt = 0, that is t = Vw, or we may 
choose to write it r = Vu/u. 

Hence, VV<r+ VVua/u = = Vv(ua), and S(ua)dp=0 
is thus proved to be exact. 

We may also proceed thus. Since every vector line is 
the intersection of two surfaces, say u = = v, then we can 
write the curl of a, which is a vector, in the form 

VV<r = hVVu\7v, 

and if S<tS7<t = 0, it follows that we must have a in the 
plane of Vw, Vfl and 

a = xVu -f- yVv. Sadp = — xdu — ydv. 

But also 

VVcr = VVxVu + VVyVv = hVVuVv. 

Hence 

SVuVyVv = = SVvVxVu. 

These are the Jacobians of u, v, x and u, v, y however, and 
since their vanishing is the condition of functional de- 
pendence, it follows that x and y are expressible as functions 
of u and v. Hence we have 

x(u, v)du + y(u, v)dv — 0. 

It is known, however, that this equation in two variables is 
always integrable by using a multiplier, say g. Therefore 
S(ga)dp = is exact for a properly chosen g. Further we 
see that ga = — Vw, or that when SaV.a = 0, a = mVw. 
If SVo- = for all points, then we find easily that 

a = Wr. 
For 

a = hVVu\/v, 



DIFFERENTIALS 193 

so that 

SV<t = SvhVuVv = 
and 

h = h(u, v). 

Integrate h partially as to u, giving 

w = fhdu + f(v), 
then 

Vw = hVu + fvVv, VX/wVv = hVX/uX/v = o\ 

Set r = wX/v or — v X7w and we have at once a = VX/j. 
It is clear that if we draw two successive surfaces W\ 
and w 2 and two successive surfaces Vi and v 2 , since 

m„ Aw , „, Av 

T\/ w = and T\7v = 



Ani An 2 

and the sides of the parallelogram which is the section of 
the tube are A<?2 = Arii esc 6, Asi = An 2 esc 6, and 
area = AniAn 2 esc 6, then TVx area = AwAv, and these 
numbers are constant for the successive surfaces, hence the 
four surfaces form a tube whose cross-section at every point 
is inversely as the intensity of a. For this reason a is said 
to be solenoidal or tubular. 

If Vx/a = for all points then we must have a = V«. 
For SvVo- = and a = gVv, VX/a = VX/gX/v, hence g 
is a function of v, and we may write 

a = X7u. 

If X7d = 0, we must have, since Sx/c = 0, a = VX/r, and 
since VX7(T = 0, a = X7u, whence X7 2 u — 0. Therefore, if 
X7(T = 0, <t is the gradient of a harmonic function and also the 
curl of a vector r, the curl of the curl of r vanishing. Also 
if VX7VX/t = 0, since we must then have Vx/t = X/v, and 
therefore SV^Vr = = V 2 fl, we can say that if the curl 



194 VECTOR CALCULUS 

of the curl of a vector vanishes it must be such that its curl 
is the gradient of a harmonic function. Also SdpVr= —dv. 
Functions related in the manner of v and r are very im- 
portant. 
Since in any case SvVVcr — 0, we must have 

Vv<r = VVuVv or VV(<r — u\/w) = 0, 

whence 

a — uVw = Vp, 

so that in any case we may break up a vector a into the form 

a = Vp + uVw. 

It follows that SaV<r = SVp\7u\/w. If we choose u, w 
and x as independent variables, we have 

Vp = PxVx + p u Vu + p w Vw, 
whence 

S<tX7(t = p x SVxVuVw, 

and we can find p from the integral 

p = fSaVv/SVxVuVw-dx. 

In case SaVcr = 0, p = constant, and a = uVw. 

A theorem due to Clebsch is useful, namely that a can 
always be put into the form 

<r = Vp + VVt, where V\/Vp = 0, SVFVr = 0, 

that is, <r can always be considered to be due to the super- 
position of a solenoidal field upon a lamellar field. We 
merely have to choose p as a solution of 

V 2 p = SVcr, 

for we have at once Sv(<r — Vp) = 0, and therefore 

o- — Vp = VVt. 



DIFFERENTIALS 195 

This may easily be seen to give us the right to set 

<r = Vp + (Vv) n r. 

EXAMPLES. SOLUTIONS OF CERTAIN DIFFERENTIAL 
FORMS 

(1). SV<t = 0, then a = VVr, and if Vv<r = 0, <r = Vp. 
If V<r = 0, <7 = Vh where V 2 ^ = 0. 

(2). If <p is a linear function dependent upon p continu- 
ously, and <pV = 0, <p = OVvQ- If <poV = 0, 

<Po = VV(6 VV0), 

8, do are linear functions. For the notation see next chapter. 

(3). VVvQ = 0, <p = - VSaQ. If e(Fv </>()) = 0, 
<P = fcFVO ~ V-SerO. If (FV^())o = <W = p() - V^(). 
Fv^o = 0, <?o = - S()V- Vp. 

(4). A particular solution of certain forms is given, as 
follows : 

*SVo" = a, cr = Jap, Fv<r = eat, a = \Vap, 
Vp = oc, p = — Sap, yXJ = ol, (p = — Sap'Q, 
VV<pQ = 6, <p = - iVpdQ, €(VV<pQ) - a, 

? = - &*p.(), (Fvrio = O , ip ; - - i^oO, 

Fw = p{), <p - - fo7p() - V&r(). 

EXERCISES 

1. Consider the cases o- = t -\-jf(g(p)) + cfc, where/ and gr have the 
following values: f = g, g 2 , g 3 , <g,fg, g~\ g~ 2 , e«, log g, sin g, tan #, and 
g has the values y/r, (y - 'ax) /(ay + »), (bx + jf)/(a; - &y), x/y, 
— x/y, — y/x, etc., V (x 2 + y 2 ) — a. 

2. Consider the vector lines of 

a = i cos (3-n-r) + j sin (3xr), r = V (x 2 + y 2 ). 

3. Consider the significance of S-Ua\/Ua = 0; give examples. 

4. If rf<r = Vt dp find F V <r. Likewise if da = adpd, da = aSpdp, da 
= —p 2 dp, da = Vradp where t is a function of p. 

17. Groups. If Si, Sj, • • • , S n are any functions linear 



196 VECTOR CALCULUS 

in V but of any degrees in p, then they form a transforma- 
tion group (Lie's) if and only if for any two Si, S;, 

where is a linear function of Si, S2, • • • S n , and a, /? 
arbitrary vectors. For instance, we have a group in the 
six formal coefficients of the two vector operators 



Si = - V - pSpV, S 2 = - FpV, 



for 



SaZiSpEi - S0Ei&*Ei = Sa(3Z 2 , 
SaE 2 S/3E 2 - £/3E 2 &*E 2 = - &x/3S 2 , 
&*SiS/3S 2 - S/SSt&xSi = - SapBi. 

The general condition may be written without a, /3 : 

Kt S E/ - Si'SZj - v e 0, 

where the accented vector is operated on by the unaccented 
one. 

Integration 
18. Definition. We define the line integral of a function 
of p,f(p), by the expression 

flf(p)<p{dp) = Lim 2f( Pi )(p(dpi), % - 1, • • •, »j 

n = 00 

where the vectors pi for the n values of i are drawn from the 
origin to n points chosen along the line from A to B along 
which the integration is to take place, <p(cr) is a function 
which is homogeneous in a and of first degree, rational or 
irrational, dpi = p t - — p z _i, and the limit must exist and be 
the same value for any method of successive subdivision 
of the line which does not leave any interval finite. Like- 
wise we define a definite integral over an area by the expres- 
sion 

ffi(p)<P2{dip,d 2 p) = Lim 2f(j>i)<to(dip it d 2 pi), 



INTEGRALS 197 

where <p 2 is a homogeneous function of dipi and d 2 p{, two 
differentials on the surface at the point pi, and of second 
degree. A definite integral throughout a volume is simi- 
larly defined by 

J % J % .ff(p)<P3(dip, d 2 p, d z p) = Lim 2/(p»)¥>g(dipt, d 2 pi, d 3 pi). 

For instance, if we consider /(p) = a, we have for ffadp 
along the straight line p = fi + #7, dp = cfo-7 and 

Lim "Zadx-y from # = # to x = Xi is 0:7(21 — Xo), 
hence 

^P = «(Pi - Po). 

The same function along the ellipse p = /3 cos + 7 sin 0, 
where dp = (— /? sin 6 + 7 cos 0)d0 has the limit 

(a/3 cos 6 -\- ay sin 0) 

between = O , 6 = 0i, that is, again a(pi — p ). 

EXAMPLES 
(1). j£« £dp/p = log TWpo, for any path. 
( 2 )- Su ~ q~ l dqq~ l = qr 1 — g _1 , for any path. 
(3). The magnetic force at the origin due to an infinite 
straight current of direction a and intensity / amperes is 
H = 0.2-I-Va/p, where p is the vector perpendicular from 
the origin to the line. In case then we have a ribbon whose 
right cross-section by a plane through the origin is any 
curve, we have the magnetic force due to the ribbon, 
expressible as a definite integral, 

H = 0.2IfVaTdp/p. 

For instance, for a segment of a straight line p = a(3 -\- xy, 
/3, 7 unit vectors Tdp = dx, 

H = 0.27 / '(ay - xt3)dx/(a 2 + x 2 ) 
= - 0.2/0 -log (a 2 + * 2 2 )/(a 2 + *i 2 ) 

-f- 0.2 -I-yitsoT 1 x 2 /a — tan -1 xj/a), 
= 0.27/3 -log OA/OB + O.27J. L AOB. 



198 VECTOR CALCULUS 

(4). Apply the preceding to the case of a skin current in a 
rectangular conductor of long enough length to be prac- 
tically infinite, for inside points, and for outside points. 

(5). Let the cross-section in (4) be a circle 

p — b3 — a(3 cos 6 — ay sin 0. 

Study the particular case when b = and the origin is the 
center. 

(6). The area of a plane curve when the origin is in the 
plane is 

\TfVpdp. 

If the curve is not closed this is the area of the sector 
made by drawing vectors to the ends of the curve. If we 
calculate the same integral \fVpdp for a curve not in the 
plane, or for an origin not in the plane of a curve we will 
call the result the areal axis of the path, or circuit. This 
term is due to Koenigs (Jour, de Math., (4) 5 (1889), 323). 
The projection of this vector on the normal to any plane, 
gives the projection of the circuit on the plane. 

(7). If a cone is immersed in a uniform pressure field 
(hydrostatic) then the resultant pressure upon its surface is 
"~ 2^Vpdp-P, where p is taken around the directrix curve. 

(8). According to the Newtonian law show that the at- 
traction of a straight segment from A to B on a unit point at 
is in the direction of the bisector of the angle AOB, 
and its intensity is 2/x sin ^AOB/c, where c is the perpen- 
dicular from to the line. 

(9). From the preceding results find the attraction of an 
infinite straight wire, thence of an infinite ribbon, and an 
infinite prism. 

(10). Find the attraction of a cylinder, thence of a solid 
cylinder. 

19. Integration by Parts. We may integrate by parts 



INTEGRALS 199 

just as in ordinary problems of calculus. For example, 

f y s V-adpSp P = iVa(B8P8 - ySfa) + \VaVPf*V pdp, 

which is found by integrating by parts and then adding to 
both sides J* y V -adpSpp. The integral is thus reduced to 
an areal integral. In case y and 5 are equal, we have an 
integral around a loop, indicated by J?. 

EXAMPLES 

(1). SfdpVcxp = HdVaS - yVay) - \Vaf*Vpdp 

+ iSafjVpdp. 

(2). f y *V.VadpV(3p = ilaSPSy'Vpdp + pS-afy'Vpdp 

- 5 Sap 5 + y Softy]. 

(3). f y *S'VadpV(3p = i(Sa8S(35 - Say Spy - 8 2 SaP 

-y 2 Sap- S-a(3f y s Vpdp). 

(4). JfV-adpVPp = U*Spf y s Vpdp + pSafjVpdp 

- dSa(38 + y Softy + Sa5S(38 

- Say Spy - 8 2 SaP + y 2 SaP 

- SaPffVpdp). 

(5). f y s SapSpdp = USadSpd - SaySPy 

-S-Voftf'Vpdpl 

(6). ffdpSap = itfSad - ySay + V-affVpdp]. 

(7). f y s Va P Spdp = HVadSpb - VaySPy - SoftffVpdp 

+ PSaJfVpdp]. 

(8). f y s Vap-dp = i[Va6'B - Vayy + af y 8 Vpdp 

+ SaffVpdp]. 

(9). fjapdp = h[a(8 2 - y 2 ) + 2af*Vpdp]. 

As an example of this formula take the scalar, and notice 
that the magnetic induction around a wire carrying a 



200 VECTOR CALCULUS 

current of value Ta amperes, for a circular path a 

B - - 2p.Vap/a 2 . 
Therefore 

- fO^Sapdp/a 2 = - SfdpB = - OSfia^SafVpdp 
= ATafia~ 2 wr 2 . 

For fj, = 1, r = a, this is OAwC. This gives the induction 
in gausses per turn. 

(10). SfSdpw - i[S8cp8 - Sy<py] + SeffVpdp. 
(11). /^prfp = h[Vy<py - V8<p8 + <p'f*Vpdp 

+ rmffVpdp]* 
(12). XVprfp = }[**.« - ^y. 7 + SeffVpdp 

- tp'f'Vpdp] - m.ffVpdp. 
For any lineolinear form 

SfQip, dp) = hm, «) - Q(y, y)] 

+ ifAQiP, dp) - Q(d P , p)} 
= ««(*, *) - Q(r t)] + WSfVpdp. 

(13). State the results for preceding 12 problems for in- 
tegration around a loop. 

(14). Consider forms of second degree in p, third degree, 
etc. 

20. Stokes* Theorem. We refer now to problem 
page 189, where we have the value of cro, a function of po, 
stated for the points in the vicinity of a given fixed point. 
If we write <tq for the value of a at a given origin 0, its 
value at a point whose vector is dp is 

o- = V 5p [- S<r 8p + %S8pVS(ro8p] - £F5pFVo% 

where V refers only to <r , and gives a value of the curl at 

* wii(v) = — Si(pi — Sj<pj — Sk<pk. For notation see Chap. IX. 






INTEGRALS 201 

the origin 0. If we multiply by ddp and take the scalar, 
we have 

Sadbp = d Sp [Sa 8p - iS8pVSa dp] + iSSpd6pVV<r . 

Therefore if we integrate this along the curve whose vector 
radius is dp we have 

ffcSed&p = [So- 8p2 - Saodpi - §S6p 2 VS<ro8p2 

+ iSSpiV Saotpi] + %SW<T fVdpd8p. 

The last expression, however, is the value of 

$[FVovareal axis of the sector between dpi and 5p 2 ]. 

Therefore for an infinitesimal circuit we have 

fSvodbp = £[FVovareal axis of circuit] = SUvVVvo-dA. 

FWo is the curl of a at some point inside the loop. If now 
we combine several circuits which we obtain by subdividing 
any area, we have for the sum of the line integrals on the 
left the line integral over the boundary curve of the area 
in question, and for the expression on the right the sum of 
the different values of the scalar of the curl of a multiplied 
into the unit normals of the areas and the areas themselves 
or the area integral ffSV\/(rdipd 2 p. That is, we have 
for any finite loop, plane or twisted, the formula 

fSadp = ffSVV(TVd lP d 2 p. 

This is called Stokes' Theorem. It is assumed in the proof 
above that there are no discontinuities of a or V\/a, 
although certain kinds of discontinuities can be present. 
The diaphragm which constitutes the area bounded by 
the loop is obviously arbitrary, if it is not deformed over 
a singularity of a or V\7a. 

It follows that fSadp along a given path is independent 
of the path when the expression on the right vanishes for 
X4 



202 VECTOR CALCULUS 

the possible loops, that is, is zero independently of dip, 
dip, or that is, V\7<r = 0. This condition is necessary and 
sufficient. 

It follows also that the surface integral of the curl of a 
vector over a diaphragm of any kind is equal to the circula- 
tion of the vector around the boundary of the diaphragm. 
That is, the flux of the curl is the circuitation around the 
boundary. 

We may generalize the theorem as follows, the expression 
on the right can be written ffSUvVVo- dA, where v is 
the normal of the surface of the diaphragm and dA is the 
area element. If now we construct a sum of any number 
of constant vectors a u a 2 , • • • a n each multiplied by a 
function of the form Saidp, Scr^dp, • • • Scr n dp, we will have 
a general rational linear vector function of dp, say <pdp, 
and arrive at the integral formula 

fvdp = ff<p(VUpV)dA, 

where the V refers now to the functions of p implied in <p. 
This is the vector generalized form of Stokes' theorem. 

If the surface is plane, Uv is a constant, say a, so that 
for plane paths 

fipdp = ff<pVVa-dA. 

We may arrive at some interesting theorems by assigning 
various values to the function <p. For instance, let 

<pdp = a dp, 
then 

<p(VUvV) = <t'VUvVv'=-Ui>SV<t+V'S<t'Uv+SUpV(t, 

whence 

ffS^a-dv = ffV'Sa'dv+ fVadp. 
If 

<fdp = pSdpa, 



INTEGRALS 203 

then 

<pVUvV = pSUvVo- - VaUv, 
therefore 

ffV-adv = ffpSdvVa - fpSadp. 
If 

ipdp = pVdpa, 
tpVUvV = pV(VUvV)<r - SUva + aVv, 

therefore 

ffvdv + Sadv = - ffpV(VUvV)<r + fpVdpa, 

hence 

2ffSadv = - ffSp(VUvV)<r + fSpdpa. 

EXERCISES 

1. Investigate the problems of article 19, page 198, as to the applica- 
tion of the theorem. 

2. Show that the theorem can be made to apply to a line which is 
not a loop by joining its ends to the origin, and after applying the 
theorem to the loop, subtracting the integrals along the radii from 
to the ends of the line, which can be expressed in terms of dx, along a line. 
Also consider cases in which the paths follow the characteristic lines of 
Vadp = 0. 

3. The theorem may be stated thus: the circulation around a path 
is the total normal flux of the curl of the vector function a through the 
loop. 

4. If the constant current la amperes flows in an infinite straight 
circuit the magnetic force H at the point p (origin on the axis) is for 

Tp<a H = ^IVa P , 
and for 

a<T P H = 0.2a?I/Va P , 

a is the radius of the wire. Then 7vH = /(a/10) inside the wire and 
equals zero outside. Integrate H around various paths and apply 
Stokes' theorem. In this case the current is a vortex field of intensity 
7ra 2 7/10. 

5. If we consider a series of loops each of which surrounds a given 
tube of vortex lines, it is clear that the circulation around such tube 
is everywhere the same. If the vector <r defines a velocity field 
which has a curl, the elementary volumes or particles are rotating, as 



204 VECTOR CALCULUS 

we have seen before, the instantaneous axis of rotation being the unit 
of the curl, and the vector lines of the curl may be compared to wires 
on which rotating beads are strung. It is known that in a perfect 
fluid whose density is either constant or a function of the pressure only, 
and subject to forces having a monodromic potential, the circulation in 
any circuit through particles moving with the fluid is constant. [Lamb, 
Hydrodynamics, p. 194.] Hence the vortex tubes moving with the fluid 
(enclosing in a given section the same particles), however small in cross- 
section, give the same integral of the curl. It follows by passing to an 
elementary tube that the vortex lines, that is, the lines of curl, move 
with the fluid, just as if the beads above were to remain always on the 
same wire, however turbulent the motion. In case the vortex lines 
return into themselves forming a vortex ring, this leads to the theorem 
in hydrodynamics that a vortex ring in a perfect fluid is indestructible. 
It is proved, too, that the same particles always stay in a vortex tube. 

6. Show that for a- = a(3S 2 ot P - 2Sp P ) + £(4# 3 /Sp - 2Sa P ), where 
Sa& = 0, the integral from the origin to 2a -J- 2/3 is independent of the 
path. Calculate it for a straight line and for a parabola. 

7. The magnetic intensity H, at the point 0, from which the vector 
p is drawn to a filament of wire carrying an infinite straight current in 
the direction a, of intensity I amperes, is given by 

H = 0.27/Fap. 

Suppose that we have a conductor of any cross-section considered as 
made up of filaments, find the total magnetic force at due to all 
the filaments. Notice that 

H = 0.2/ Fa V log TVap, 

and that a is the unit normal of the plane cross-section of the conductor. 
Hence 

ffHdA = ff0.2IVaV log TVapdA = f0.2I log TVapdp 

around the boundary of the cross-section. This can easily be reduced 
to the ordinary form 0.21 j? log rdp. This expression is called a log- 
arithmic potential. If I were a function of the position of the filament 
in the cross-section, the form of the line-integral would change. 

For a circular section we have the results used in problem 4. Con- 
sider also a rectangular bar, for inside points and also for outside points. 

8. If or and r are two vector functions of p, we have the theorem 

SVUuVVo-t = St(VUvV)* - S<t(VUpV)t, 
whence 

ffSr(VUpV)o- = ffS<r{VUvV)r + fSdpar, 



INTEGRALS 205 

for a closed circuit. Show applications when a or t or both are sole- 
noidal. 

9. Show that 

ffS-dvotS\7<x = fSdpaa + ffSdv(SaV)<r, 
ffS-Vuadv = JTuSadp - f fuSV adp, 
ffS-X7uS7vdv= fuSsjvdp = - fvS\7ud P , 
f 1 hiSVvd P = [uv] p p l - f^vSS/udp. 

10. Prove Koenig's theorems and generalize. 

(1) Any area bounded by a loop generates by translation a volume 
= — Saw, where co is the areal axis; 

(2) The area for a rotation given by (a + Vap)at is — J] Saco + 
f t Scf VpVpdp. 

21. Green's Theorem. The following theorem becomes 
fundamental in the treatment of surface integrals. Refer- 
ring to the second form in example 17, page 189, for the 
expression of a vector in the vicinity of a point, which is 

0" = FV Sp [- iV8pa + iS8pVVdp<ro] - l&pSV<To 

we see that if we multiply by a vector element of surface, 
Vdi8pd 2 dp, and take the scalar 

Scdrfpdidp = SUvVs p []dA - iSV(r Sd l dpd 2 dp8p. 

If now we integrate over any closed surface the first term 
on the right gives zero, since the bounding curve has be- 
come a mere point, and thus, indicating integration over 
a closed surface by two J', 

j> $&<jd\hpd<ihp = — \S\7(TQjf jfSdibpd'ibpbp. 

But the last part of the right hand member is the volume 
of an elementary triangular pyramid whose base is given by 
didpd 2 8p. Hence, the integral is the elementary volume of 
the closed surface, and may be written dv, so that we have 
for an elementary closed surface 

j> \fSad18pd2dp = SVvodv. 



206 VECTOR CALCULUS 

If now we can dissect any volume into elements in which 
the function has no singularities and sum the entire figure, 
then pass to the limit as usual, we have the important 
theorem 

ffS<rd lP d 2 p = fffSVv dv. 

This is called Greens theorem, or sometimes Green's theorem 
in the first form. It is usually called Gauss' theorem by 
German writers, although Gauss' theorem proper was only 
a particular case and Green's publication antedates Gauss' 
by several years. 

The theorem may be stated thus: the convergence of a 
vector throughout a given volume is the flux through the 
bounding surface. 

It is evident that we can generalize this theorem as we did 
Stokes' and thus arrive at the generalized Green's theorem 
$ fQvdA = f f f$\/ dv. v is the outward unit normal. 

The applications are so numerous and so important that 
they will occupy a considerable space. 

• The elementary areas and volumes used in proving Stokes' 
and Green's theorems are often used as integral definitions 
of convergence or its negative, the divergence, and of curl, 
rotation, or vortex. For such methods of approach see Joly, 
Burali-Forti and Marcolongo, and various German texts. 

A very obvious corollary is that if SVc = then 

$ \fSad1pd2p = 0. 

It follows that the flux of any curl through any closed sur- 
face is zero. Hence, if the particles of a vortex enter a 
closed boundary, they must leave it. Therefore, vortex 
tubes must be either closed or terminate on the boundary 
wall of the medium in which the vortex is, or else wind 
about infinitely. We may also state that if SVa = the 
differential expression Sadipd 2 p is exact in the sense that 



INTEGRALS 207 

J % J % S(rdipd 2 p is invariant for different diaphragms bounded 
by a closed curve, noting the usual restrictions due to 
singularities. 

We proceed to develop some theorems that follow from 
Green's theorem. Let $Uv be — pSUvcr, then 

3>V = — pSv<r + o- 
and we have 

fffadv = fffpSVvdv - ffpSUvadA. 

Let $Uv = — pVUva, then <i>V = — pVVv + 2a and 
SSfvdv = ifffpVVvdv - \ffpVUvodA. 

Let $Uv = pSpUva, then <J>V = pSpVv + Fpo-, whence 
fffVpa dv = - fffp&Va dv + ffpSpUvadA. 

Let $17V = - pVpVUixr, then $V = - pFpFVo" + 3PV, 

hence 

SSSVpadv = ifffpVpWadv- \ffpVpVUvadA. 

Let $E7V = SprUiHT, then 3>V = SprV<r + Spa\/r + Sot, 

thence 

fffSar dv = - fffiSprV* + &rVr)<fo 

+ f f&prTJva dA. 

In the first of these if a- has no convergence we have the 
theorem that the integral of cr, a solenoidal vector, through- 
out a volume is equal to the integral over the surface of p 
multiplied by the normal component of a. In the second 
we have the theorem that if the curl of a vanishes through- 
out a volume, so that a- is lamellar in the volume, then the 
integral of a throughout the volume is half the integral 
over the surface of p times the tangential component of a 
taken at right angles to a-. In the third, if the curl of cr 



208 VECTOR CALCULUS 

vanishes then the integral of the moment of a with regard 
to the origin is the integral over the surface of Tp 2 times the 
component along p of the negative of the tangential com- 
ponent of a taken perpendicular to <r, and by the fourth 
this also equals the surface integral of the component 
perpendicular to p of the negative tangential component of 
<r taken perpendicular to a. According to the fifth formula, 
if a solenoidal vector is multiplied by another and the scalar 
of the product is integrated throughout a volume, then the 
integral is the integral of — SpaVr throughout the volume 
-f- the integral of ScrprUv over the surface. 

If in the first, second, third, and fourth we set c<t for a, 
and in the fifth ca for a and — \<t for r, we have from the 
first and second the expression for X, the momentum of a 
moving mass of continuous medium, of density c, and from 
the third and fourth the moment of momentum, /x, and 
from the fifth the kinetic energy. If the medium is in- 
compressible, and we set 2k = V\/v, since SVca = 0, then 

X = fffcadv 
= - ffcpSUvadA + fffpSaVcdv 

+ SSfcpSV* dv 

= fffpcKdv+lfffpWcadv - \££cpVVvadA. 

ju = fffcVpadv 
= ffcpSpUvadA - SSfcpSpVa - fffpSpVca 
= UffcpVpK + \fffpVpWcadv 

- \ffcpVpVVvadA. 
T = - hSfSSa 2 cdv 

= - hffSpvUvac dA + SffhcSpaVo- dv 

+ hfffSpvVca dv. 

In case c is uniform these become still simpler. 

If we set a = S/u and r = \/w in the above formula we 



INTEGRALS 209 

arrive at others for the gradients of scalar functions. The 
curls will vanish. If further we suppose that u, or w, or 
both, are harmonic so that the convergences also vanish 
we have a number of useful theorems. 

Othei forms of Green's theorem are found by the follow- 
ing methods. Set $Uv = uS\7wUv, then 

$V = u\/ 2 w + SVuVw 

and we have the second form of Green's theorem at once 

SfS&VuVw dv = ffuS\/wUv dA — fffu\7 2 wdv, 

and from symmetry 

yWSvWw dv = ffwSVuUv dA — fffw\/ 2 u dv. 

Subtracting we have 

J % J *J % (u\7 2 w — w\7 2 u) dv 

= ~ f£(,STJv[u\7w - wVu])dA. 

22. Applications. In the first of these let u = 1, then 
fffV 2 ™ dv — — J'.fSUj'VwdA. If then w is a har- 
monic function, the surface integral will vanish, and if V 2 w 
= 47Tju, which is Poisson's equation for potentials of forces 
varying as the inverse square of the distance, inside the 
masses, ju being the density of the distribution, then 

ffSUvS7w dA = ±ttM, 

where M is the total mass of the volume distribution. This 
is Gauss' theorem, a particular case of Green's. In words, 
the surface integral of the normal component of the force 
is — 47r times the enclosed mass. The total mass is l/4x 
times the volume integral of the concentration. 

In the first formula let u = 1/Tp and exclude the origin 



210 VECTOR CALCULUS 

(a point of discontinuity) by a small sphere, then we have 

fffSV(l/Tp)Vwdv 

= ffdA SUrVw/T P - fffdv V 2 w/Tp 

for the space between the sphere and the bounding surface 
of the distribution w, and over the two surfaces, the normals 
pointing out of the enclosed space. But for a sphere we 
have dA = Tp 2 dw where co is the solid angle at the center, 
and dv = Tp 2 dwdTp. Thus we have 

fffV 2 w/Tp dv 

= ffSdA UuVw/Tp -fffSv(l/T P )Vwdv 
= ffSdA UvVw/Tp -fffSv(wV[l/T P ])dv 

since V 2 l/7p = 0, 
= ffSdA UvVwjTp -ffSdA UvwV(l/Tp) 
= ffSdA UvVwjTp+ffSdA VvVp\T 2 pw. 

Now of the integrals on the right let us consider first the 
surface of the sphere, of small radius Tp. The first integral 
is then - ffSUpX/wlTp- T 2 pdco = - ffSUpVw- Tpda, 
and if we suppose that the normal component of Vw, that 
is, the component of Vw along p, is everywhere finite, then 
this integral will vanish with Tp. The second integral for 
the sphere is — J?rf'SUpUpwT 2 pd(x)lT 2 p = — tfj'wdu, 
and the value of w at the origin is Wo, then this integral is 
47TWo since the total solid angle around a point is 47r. 
Hence we have 

fffdv V 2 w/Tp = ffSUv{\/wlTp + wUp/T 2 p)dA 

+ 4twq 
and 

4x^o= fffdvV 2 w/Tp 

- ffSUp(Vw/Tp -f wUp/T 2 p) dA, 



INTEGRALS 211 

where the volume integral is over all the space at which w 
exists, the origin excluded, and the surface integral is over 
the bounding surface or surfaces. In words, if we know the 
value of the concentration of w at every point of space, 
and the value of the gradient of w and of w at every point 
of the bounding surfaces at which there is discontinuity, 
then we can find w itself at every point of space, provided 
w is finite with its gradient. If X7 2 w is of order in p not 
lower than — 1 we do not need to exclude the origin, for 
the integral is ///V 2 ^ TpdcpdTp, and this will vanish 
with Tp when V 2 w is not lower in degree than — 1. 

EXERCISES 

1. We shall examine in detail the problem of w — const, over a given 
surface, zero over the infinite sphere, V 2 w = everywhere, \/w = 
on the inside of the sphere, but not zero on the outside. Then for the 
inside of the sphere the equation reduces to 

4:irw = - £fwSUvUplT*pdA = 4ttu;, 

hence w is constant throughout the sphere and equal to the surface value. 
On the outside of the sphere, we have to consider the bounding sur- 
faces to be the sphere and the sphere of infinite radius, so that we have 

4^0 = _ ffSdA UvVw/Tp- wffSdA UuUpfTp 2 , 

where the first integral is taken over both surfaces and the second 
integral is over the given surface only, since w = at °° . The second 
integral vanishes, however, since it is equal to w times the solid angle 
of the closed surface at a point exterior to it. If we suppose then that 
\/w is at «3 we have a single integral to evaluate 

4:irw = — j> j> 'SdAU r i>\? 'w/T 'p over the surface. 

A simple case is 

— SUv\/w = const. = C. 
Then 

4ttWo = CffdAITp. 

The integration of this and of the forms arising from a different assump- 
tion as to the normal component of V^ can be effected by the use of 
fundamental functions proper to the problem and determined by the 
boundary conditions, such as Fourier's series, spherical harmonics, 
and the like. One very simple case is that of the sphere. If we take 



212 VECTOR CALCULUS 

the origin at the center of the sphere we have to find the integral 

,f,fdA/T( P - P o) 

where po is the vector to the point. Now the solid angle subtended 
by po is given by the integral — r~ l ffdASpU{p — po)/T*(p — p ) 
and equals 4t or 0, according as the point is inside or outside of the 
sphere. This integral, however, breaks up easily into two over the 
surface, the integrands being 

r-^T-Kp - po) - SpoU( P - P0 )/T*(p - po), 

but the last term gives or — 47rr 2 /7 T p , as the point is inside or outside 
of the sphere. Hence the other term gives 

ffdAlT{p - po) - 47rr or 4Trr 2 /Tp 

as the point is inside or outside. We find then in this case that 

w m Cr 2 /Tpo. 

If in place of the law above for — SUvS7w, it is equal to C/T 2 (p — p ) 
we find that 

ffdAIT\ P - po) = 47rr/(r' + p 2 ) 
or 

47^/(7^0 - r^po). 
Inside 

_ r r ,A S(p ~ pp)(p + po) 

- ffdA TKp - po) ' 

dA = 27rr 2 sin Odd =- d[a 2 + r 2 - x 2 ] = —xdx, 
a a 

„ po(p — po) = ax cos 4/ _ a 2 + x 2 — r 2 
T*(p - po) " x 3 2x* 

ffdAS^f^=^f r+a ' a+r ( a ^ + l)dx = 

T 2 (p — po) aJr-a,a-r \ X 2 J 

or 

47TT 2 

a 

The differentiation of these integrals by using Vp as operator under 
the sign leads to some vector integrals over the surface of the sphere. 
2. Show that we have 

££UvdAIT(p - po) = |ttpo or |7rr 3 /^ 3 Po-po 

for inside or outside points of a sphere. 



INTEGRALS 213 

3. Find ffdAUu/T 3 ( P - Po ) for the sphere. 

4. Prove f fdAT^{p-fi)T-\p-oc) =4 7 rr/[(r 2 -a 2 )7 7 ( / S-«)] or 

= ^r 2 J[a(r 2 -a 2 )T(r 2 a- 1 +0)]. 

5. Consider the case in which the value of w is zero on a surface 
not at infinity but surrounding the first given surface. We have an 
example in two concentric spheres which form a condenser. On the 
inner sphere let w be const. = Wi, on the outer let w = 0, on the inner 
let — SUpVw = 0, inside, = E h outside, on the outer let — SUv\/w 
= E 2 on the inside, = Oon the outside. 

6. If w is considered with regard to one of its level surfaces, it is 
constant on the surface, and the integral — £ f SdAU vU p\T 2 pio 
becomes for any inside point 4:irw, hence we have 

4irw - A.™ = fffdv\7 2 wlT P - £ £SdAUuVw/T P . 

If then w is harmonic inside the level surface, it is constant at all points 
and 

47r(w - to) m - £fSdAUv\7wlTp. 

But since w is constant as we approach the surface, V^o =0, and 
V(w — Wo) = 0, so that X7w = 0. Hence w = w. If w vanishes at 
oo and is everywhere harmonic it equals zero. 

7. If two functions Wi, w 2 are harmonic without a given surface, 
vanish at » , and have on the surface values which are constantly in the 
ratio X : 1, X a constant, then W\ = \W2. 

8. If the surface Si is a level for both the functions u and w, as also 
the surface S 2 inside Si, and if between Si and $2, u and w are harmonic, 
then 

(U — Ui)(w 2 — Wi) = (W — Wi)(ll2 — Ui). 

For if w = <p(u), then V 2 w = = <p"(u)T 2 \7u, hence <p(u) — au + b, 
etc. 

[A scalar point function w is expressible as a function of another 
scalar function u if and only if V\/w\7u = 0.] 

9. Outside a closed surface S, Wi and w 2 are harmonic and have the 
same levels. Si vanishes at • while w 2 has at 00 everywhere the con- 
stant value C. Then w 2 = Bwi + C. 

For Vw 2 = tVw h V 2 w 2 = V^V^i = 0, thus V* = 0, or V^i = 0, 
and t = B or wi = const. 

10. There cannot be two different functions W\, w 2 both of which 
within a given closed surface are harmonic, are continuous with their 
gradients, are either equal at every point of S or else SUvX/Wi =SUv\/w 2 
at every point of S while at one point they are equal. 

Let u = Wi — w 2 , then V 2 w = 0, SJu = on S or else SUv\/u = 0, 
and at one point Vw = 0. 



214 VECTOR CALCULUS 

11. Given a set of mutually exclusive surfaces, then there cannot be 
two unequal functions w\, Wi, which outside all these surfaces are 
harmonic, continuous with their gradients, vanish at <» as Tp~ l , their 
gradients vanishing as Tp -2 , and at every point of the surfaces either 
equal or SUvVwi = SUvVwt- 

23. Solution of Laplace's Equation. The last problems 
in the preceding application show that if we wish to invert 
V 2 w = 0, all we need are the boundary conditions, in order 
to have a unique solution. In case V 2 u is a function of 
P>f(p)> we can proceed by the method of integral equations 
to arrive at the integral. However the integral is express- 
ible in the form of a definite integral, as well as a series, 

w = l/4:w[fSSdvV 2 w/Tp 

- ffSUviVw/Tp + wUp/T 2 p)dAl 

The first of these integrals is called the potential and written 
Pot. Thus for any function of p whatever we have 

Vot q, = fffqdvlT(p- p Q ) 

where p describes the volume and p is the point for which 
Pot qo is desired. Let Vo be used to indicate operation as 
to po, then we have 

Vo Pot g = VoffSqdv/T(p - p ) 

= fff[dvU(p - p )/r 2 (p - Po )]q 
- -SSfV[qlT(p- p )]dv 

+ SffdWq/T(p- po) 
= Pot Vg - ffdAUvqlT(p - Po ). 

If we operate by Vo again, we have 

Vo 2 Pot q = Pot V 2 ? - ffdA[Uv\7qlT(p - po) 

+ V'Uvq/T'(p - po)]. 

But the expression on the right is 4x^0, hence we have the 



INTEGRALS 215 

important theorem 

Vo 2 Pot q = 4:irq . 

That is, the concentration of a potential is 4x times the 
function of which we have the potential. In the case of a 
material distribution of attracting matter, this is Poisson's 
equation, stating that the concentration of the potential 
of the density is 4r times the density; that is, given a 
distribution of attracting masses, they have a potential at 
any given point, and the concentration of this potential at 
that point is the density at the point -5- 4-7T. 

The gradient of Pot q was called by Gibbs the Newtonian 
of g , when the function q is a scalar, and if q is a vector, 
then the curl of its potential is called the Laplacian, and the 
convergence of its potential is called the Maxwellian of q . 
Thus 

New q = Vo Pot P, Lap <r = V\/o Pot <r , 
Max (To = £ Vo Pot co. 

We have the general inversion formula 

47rVo~ 2 Vo 2 ? = 47rgo 

- SSfV 2 q/T(p - Po )dv 

- ffdA[UvTqlT{p - p ) 

+ U(p - p )qUr/T*(p - p )J. 

This gives us the inverse of the concentration as a potential, 
plus certain functions arising from the boundary conditions. 
We may also define an integral, sometimes useful, called 
the Helmholtzian, 

Him. Q m fffQT{p - Po )dv. 

Certain double triple integrals have been defined: 

Pot 0, v) = ffffffu(p 1 )v(p2)dv 1 dv 2 /T(p 1 - p 2 ), 



216 VECTOR CALCULUS 

Pot (to) = fffffS - Sh dvidvJTfa - p 2 ), 
Lap (to) = ffffff + 5to(Pi - p 2 )^i^ 2 /P( Pl -p 2 ), 
New («, f) = SSSSSS-S{i(pi-p2)v l dvidv 2 IT'(pi-p t ), 
Max(£,*) = - ffffffv l SUpi-P2)dv l dv 2 ir( Pl - P2 ). 

EXERCISES 

1. Iff = — VP is a field of force or velocity or other vector arising 
from a scalar function P as its gradient, then 

Po = - SSSSV£dv/[4irT( P - po)] + ffdA[SUvll&*T{p - po)) 

+ PC/ y V^(p-po)/47r]. 

If P is harmonic the first term vanishes, if £ = the first two vanish. 

2. If £ = V<r, that is, it is a curl of a solenoidal vector, 

°o = fffVV* dv/[4irT(p - po)] - f<fdA[VUv<rl[±TcT{p - po)] 

+ U(p - p 9 )<rlU,[4*T*(j> - po)]. 

3. We may, therefore, break up (in an infinity of ways) any vector 
into two parts, one solenoidal and the other lamellar. 

Thus, let a = 7T + t where £v r ■ 0, and Wir = 0, then Sv <r = SVx 
and since VVt = 0, this may be written Vt = &Vo" whence x. 

VVc = FVt = Vr whence t. We have, therefore, from these two 
47T<r = VfffSdvV<r/T(p - Po ) - V £ £SdAUvalT{p - Po ) 

+ V jffPSdA UvV (UT(p - po) + V V SSfVV*dv/T(p - Po ) 
-WffVU*adA/T(p-po) + VVffDSU*S7(p+( P -po)dA t 

where P is such that V 2 P = *S'V<r and D such that \7 2 D = Vs/a. 

3. Another application is found in the second form of Green's 
theorem. According to the formula 

SffiyW 1 ™ - wV 2 u) dv = - tf£(SUv[u\7w - w\/u])dA 

it is evident that if G is a function such that V 2 G = 0, and if, further, 
G has been chosen so as to satisfy the boundary condition SUvS7G = 0, 
then the formula becomes 

SffGs^wdv = - ££SUv\7wGdA. 

If then V 2 w is a given function we have the integral equation 

JfGSUvVwdA - - fffGj{ P )dv. 

Similar considerations enable us to handle other problems. 

4. If u and w both satisfy V 2 / = 0, then we have Green's Reciprocal 
Theorem: 

ffuSUvSfw dA = ffwSUvVudA, 



Thus let 
therefore 



INTEGRALS 217 






ff ^p dA = ffuSUvV -L dA. 



5. Let A relate to a as V to p ; then 

A Pot Q = ff/QdvU(p - a)/T*( P - a) 

= fffV(Q/T(p - cc))dv + fffdWQ/T(p - a) 

= Pot VQ - ffdAUuQ/T( P - a). 

If Q — on the surface, the surface integral = 0. 

New P = Pot V - ffVvP dAjT{p - a) = A Pot P when Pot exists. 

Lap a = V Pot Vo- - ££VTJvadA\T{p - a) = VA Pot a when Pot 
exists. 

Max o- = S Pot V<r - £ fSUvadAITip -a) = SA Pot o- when Pot 
exists. 

A 2 PotQ = Pot V 2 Q - ££U v \7QdAIT(p - a) 

+ //diVi[^/7 7 i( P -«)]. 
If Q = on the surface, that is, if Q has no surface of discontinuity, 

A 2 Pot Q - Pot V 2 Q, 

A New P = A 2 Pot P, 

A Lap o- = A7A Pot a, 

A Max a = A/SA Pot <r. 

6. If j8 is a function of the time t, then 



d--^[yy/i(rvFv^+M^J 



t+br 



r r 

+ VV £<fj Vdu p t+br - ff Vdv WPt+br 

where the subscript means t + br is put for t after the operations on 
have occurred. 



15 



CHAPTER IX 

THE LINEAR VECTOR FUNCTION 

1. Definition. If there is a vector a which is an integral 
rational function <p of the vector p, 

a = <p' P , 

and if in this function we substitute for p a scalar multiple 
tp of p, then we call the vector function a linear vector func- 
tion if a becomes ta under this substitution. It is also called 
a dyadic. 

The function <p transforms the vector p, which may be 
in any direction, into the vector <r, which may not in every 
case be able to take all directions. If p = a, then we have 
(pp = <pa, and <p as an operator has a value at every point 
in space. We may, in fact, look upon <p as a space trans- 
formation that deforms space by a shift in its points leaving 
invariant the origin and the surface at infinity. In the 
case of a straight line 

Vap = /?, or p = xa + cT l (5, 

we see that the operation of <p on all its vectors gives 

a = x<pa + (pVa~ 1 ^ f 

and this is a straight line whose equation is 

Vipaa = V<pa(pVa~ 1 ^, 

which will later be shown to reduce to a function of (3 
only, <p(3. Hence <p converts straight lines into straight 
lines. The lines a for which Vacpa = 0, remain parallel 
218 



THE LINEAR VECTOR FUNCTION 219 

to their original direction, others change direction. Again 
if we consider the plane S-afip = or 

p = xa + y(3, , a = x<pa + y<pf}, 

so that 

S(r<pct<pp = 0. 

Hence planes through the origin, and likewise all planes, 
are converted into planes. These will be parallel to their 
original direction if Va(3 = uV<pa<p(3, or 

VVa$V(pa<p& = = Scx<pa<p(3= S(3<pa<p(3= Sa(3<pa = So@<pP. 

Now Va(3 is normal to the plane, and /3 is any vector in the 
plane, and <p(3 by the equation is normal to Vafi, hence 
<p(3 = va + w(3 for all vectors in the plane. 

Since <p0 = 0, the function leaves the origin invariant. 
Consequently the lines and planes through the origin that 
remain parallel to themselves are invariant as lines and 
planes. These lines we will call the invariant lines of <p, 
and the planes the invariant planes of tp. 

2. Invariant Lines. In order to ascertain what lines are 
invariant we solve the equation 

Va<pa = 0, or (pa = ga, 

that is 

(tp - g)a = 0. 

First we write a in the form 

aS\fiv = \SfJiva + ixSvka + vSXfxa, 

where X, ju, v are any three noncoplanar vectors. Then we 
have at once 

(<p — g)\Sixvicx + (<p — g)nSv\a + (<p — g)pS\fxa = 0. 



220 VECTOR CALCULUS 

But this means that we must have for any three non- 
coplanar vectors X, /i, v 

S(<p - g)\(<p - g)fi(<p - g)v = 

= tfSXiiv — g 2 (S\ii<pv + S\<ptxi> + S<p\nv) 

+ g(S\(pfJL(pV + S\jJl<pV + S(f\(pfJLP) — S<p\<piJ.<pi>, 

an equation to determine g, which we shall write 

g z - mig 2 + m 2 g - m 3 = 0, 
called the /a<6n< equation of #>, where we have set 

Wl = (S\jA<pV + S\<pflP + S<p\fAl>)/S\fJLl>, 

rri2 = (S\(pii(pp -+- S<pkyupv + S(p\<piJLp)lS\fjLv, 

These expressions are called the nonrotational scalar in- 
variants of <p. That they are invariant is easily seen by 
substituting X' + v/jl for X. The resulting form is precisely 
the same for X r , ju, p, and from the symmetry involved this 
means that for X, /x, v we can substitute any other three 
noncoplanar vectors, and arrive at the same values for 
mi, m2, m 3 . It is obvious that m 3 is the ratio in which the 
volume of the parallelepiped X, jjl, v is altered. If m 3 = 
one or more of the roots of the cubic are zero. The number 
of zero roots is called the vacuity of (p. If is obvious that 
the latent cubic has either one or three real roots. 

3. General Equation. We prove now a fundamental 
equation due to Hamilton. Starting with <p we iterate the 
function on any vector, as p, writing the successive results 
thus 

p, <pp, <p<pp = <p 2 p, <p<p<pp = <p<p 2 p = <p 3 p, "•. 

We have then for any three vectors X, ,u, v that are not 
coplanar 



THE LINEAR VECTOR FUNCTION 221 

S\pi>(<p 3 p — mi<p 2 p) = (p 2 (<pp — m\p)S\pv 

= <p 2 [<p\Spvp -\- • - - — pSpv(p\ — • • •] 

= - <p 2 [V-VppV<p\p+ •••] 

= <p\V'V<p\pVp.v+ •••] 

= <p[<p\Sv<ppp + <pp<S\<pvp + <pvSp<p\p 

— <p\SpL(pvp — (fpSvcpXp 

— <pi>S\(pfxp]. 
Adding to this result S\pu> -m%ipp, we have 

S\pv((p 3 p — mnp 2 p + m<npp) 

= <p[\S<pfi<pvp + pSipVipkp + vS<p\(pp,p] = pS(f\cppapv. 

Subtracting SXfMV-rritp from both sides and dropping the 
nonvanishing factor S\p,i>, we have the Hamilton cubic for <p 

<p s p — mi<p 2 p + m*<pp — mzp = 0. 

This cubic holds for all vectors p, and hence, may be written 
symbolically 

<p 3 — mnp 2 + m 2 (p — ra 3 = 

identically. This is also called the general equation for <p. 
It is the same equation so far as form goes as the latent 
equation. Hence we may write it in the form 

(<p - gi)(<p — g*)(<p — gz) = 0. 

In other words, the successive application of these three 
operators to any vector will identically annul it. 

We scarcely need to mention that the three operators 
written here are commutative and associative, since this 
follows at once from the definition of linear vector operator, 
and of its powers. 

It is to be noted, too, that <p may satisfy an equation of 
lower degree. This, in case there is one, will be called the 
characteristic equation of <p. Since <p must satisfy its general 



222 VECTOR CALCULUS 

equation, the process of highest common divisor applied 
to the two will give us an equation which <p satisfies also, 
and as this cannot by hypothesis be lower than the char- 
acteristic equation in degree and must divide it, it is the 
characteristic equation. Hence the factors of the char- 
acteristic equation are included among those of the general 
equation. We proceed now to prove that the general 
equation can have no factors different from the factors 
of the characteristic equation. 

(1) Let the characteristic equation be 

(<p - g)p = 

for every vector; then assuming any X, /x, v, we find easily 
for the latent equation 

x*-Sgx 2 +3g 2 x-g*= 0, 

so that the general equation is 

(cp - gf = 0. 
In this case 

if = [g\SM) + gpSrkQ + gpSlnOV&V, 

where X, /z, v are given for a given <p. 

(2) Let the characteristic equation be 

(<P - 9i)(<P - 92) P = 0, 

then by hypothesis, there is at least one vector a for which 
we have 

(<p - gi)a + 0, 

and at least one fi for which 

(<p - gt)0 4= 0. 
Let us take then 

O - gi)a = X, (<p— g 2 )(S = M- 



THE LINEAR VECTOR FUNCTION 223 

Then 

(<p - g 2 )\ = 0, (<p- gi)fx = 0. 

Hence, we cannot have X and ju parallel, else gi = g 2 , which 
we assume is not the case, since from 

(<p- g 2 )U\ = 0, (<p- g 1 )U f x= 0, 
we have 

g 2 U\ = giUn, and g 2 = g u 

if X is parallel to /z, that is if U\ would = Up, 

There is still a third direction independent of X and /z, 
say v. Let 

cpv = av + bjjL + cX. 
Then we have 

(<p - ft)* = (a - gi)j>+ bfx + cX. 
Since 

(<p- fc)(* - 9i)v = 0, 
(a - gx)(<p - g 2 )v — b(g 2 - fi)p = 

= (a— g Y ){a — g 2 )v + 6 (a - g 2 )fx + c(a — g{)\. 

We must have, therefore, either 

a = gi and 6=0, 
or 

a = g 2 and c = 0. 

As the numbering of the roots is immaterial, let us take 
a = gi t b = 0, then 

<pv = giv + cX, <pX = # 2 X, ^>m = 9iV> 

We notice that if c # 0, we can choose v' = v — (cjg 2 )\, 
whence ipv' = giv' and we could therefore take c = 0. 
Hence 

g 3 - g\2gi + g 2 ) + ^(2fir^ 2 + g Y 2 ) - g?g 2 = 0, 
<p = [guiS\vQ + givSXpQ + # 2 X£mK)]ASX/xj>, 



221 VECTOR CALCULUS 

and the general equation is 

(<P - 9i) 2 (<P ~ 92) = 0. 
(3) Let the characteristic equation be 
(<p - g)*p = 0. 
Then there is one direction X for which we have 

<p\ = g\, 

and there may be other directions for which the same is 
true. There is at least one direction \i such that 

(cp - g)fi = X. 
We have, therefore, 

<PV = g» + X <?X = g\. 
Let now v be a third independent direction, then we have 

(pv = av + bjj. + cK, 
(<p - g)v = (a — g) v + 6/x + c\, 
(<p - gfv = = (a - gfv + b(a - g)p + [b + c(a - g)]K. 

Therefore, we have a = g, 6=0, <pp = gv + cX and 
<£>(*> — c/x) = g{y — c/jl) = gv' , and the general equation 

i* - g) 3 = 0, 

<p = g + XSj/XO/SX/x*'. 

We are now in a position to say that the general equation 
has exactly the same factors as the characteristic equation. 
Further we can state as a theorem the following: 

(a) // the characteristic equation is of first degree, 

O - g Y )p = 0, 

then every vector is converted into g\ times that vector, by the 
operation of (p. 



THE LINEAR VECTOR FUNCTION 225 

(6) // the characteristic equation is of the form 
O - 9i)(<P ~ 92) = 0, 

then there is one direction X such that <pk = 92K, while for 
every vector in a given plane of the form x\i.-\- yv we have 

(<p- #i)Om + yv) = 0. 

Hence <p multiplies by gi every vector in the plane of /a, v, 
and by g 2 all vectors in the direction X. 
(c) If the characteristic equation is 

W - g,f = 0, 

there is a direction such that 

<p\ = gi\ 

and a given plane such that for every vector in it x\x-\- yv 
we have 

(<P — 9i)(w + yv) = ^X. 

If (<P — gi)v = v\ (<P — gi)v = w\ we may set 

• 
w 

giving (p/i = gip. Therefore <p extends all vectors in the ratio 
gi, and shears all components parallel to v in the direction X. 
4. Nondegenerate Equations. We have left to consider 
the three cases 

(<p — 9i)(<p — 92) (<p - gz) = 0, 

O - gi) 2 (<P - 02) = 0, 

(v - g,f = 0. 

In the last case we see easily that there is a set of unit 
vectors X, ju, v such that 



226 VECTOR CALCULUS 

<p\ — g{K + mo, 

<PH = giii + vb, 

<pv = giv. 
Hence we see that 

<p(x\ + yi* + zv) = gi{x\ + y\x + zv) + a*M + 6y*> 
= gi(x\ + 2/M + •*) + a(a*M + 0*0 

+ (6 - a)yy, 
<p(x» + yv) = gi(xn + ?/*>) + fo^, 

<p = gi + [apSuvQ + bvSv\Q]/S\nv. 

Therefore <p extends all vectors in the ratio g\, shears all 
vectors X in the direction of m> and all vectors /x in the 
direction v. 

In the first case we see that there is at least one vector 
p such that 

{<P - 9\){<P - 9s) P = A, 
where 

<p\ = g{K. 

Likewise there are vectors that lead to /x and v where 
<PH = g 2 n, <pp = gzv. These are independent, and there- 
fore if we consider any vector 

p = x\ + yn + zp, 
we have 

<pp = xg{K + 2/02M + zg 3 v, 
<p = [g^SfivQ + fwJSrhO + gsvS\nO]lS\fiP. 

Evidently we can find X, /x, v by operating on all vectors 
necessary in order to arrive at nonvanishing results by 

(<P — 9z)(<P — 9s), (<P — 9i)(<P — 9*)> (<P — 9\)(<P — 9t) 

respectively. 

In the second case, we see in a similar manner that there 



THE LINEAR VECTOR FUNCTION 227 

are three vectors such that 

£>X = g{K + \x, <pfi = giii, <pv = g 2 p, 
<P = IgiQiSpvQ + nSvkQ + fiwStoQ + jtSMW&Vv. 

5. Summary. We may now summarize these results in 
the following theorem, which is of highest importance. 

Every linear vector function satisfies a general cubic, and 
may also satisfy an equation of lower degree called the char- 
acteristic equation. If the equation of lowest degree is the 
cubic, then it may have three distinct latent roots, in which 
case there corresponds to each root a distinct invariant line 
through the origin, any vector in each of the three directions 
being extended in a given ratio equal to the corresponding root; 
or it may have two equal roots, in which case there corresponds 
to the unequal root an invariant line, and to the multiple root 
an invariant plane containing an invariant line, every vector 
in the plane being multiplied by the root and then affected by 
a shear of its points parallel to the invariant line in the plane; 
or there may be three equal roots, in which case there is an 
invariant line, a plane through this line, every line of the 
plane through the origin being multiplied by the root and its 
points sheared parallel to the invariant line, and finally every 
line in space not in this plane is multiplied by the root and 
its points sheared parallel to the invariant plane. In case 
the function satisfies a reduced equation which is a quadratic, 
this quadratic may have unequal roots, in which case there 
is an invariant line corresponding to one root and an invariant 
plane corresponding to the other, any line in the plane through 
the origin being multiplied by the corresponding root; or there 
may be two equal roots, in which case there is an invariant 
plane such that every line in the plane is multiplied by the 
root and every vector not in the plane is multiplied by the root 
and its points displaced parallel to an invariant line. In case 



22$ VECTOR CALCULUS 

the reduced equation is of the first degree, every line is an 
invariant line, all vectors being extended in a fixed ratio. 
Where there are displacements, they are proportional to the 
distance from the origin, and the region displaced is called a 
shear region. 

Hence <p takes the following forms in which g if g 2 , gz may- 
be equal, or any two may be equal: 

I. [g&SpyQ + g 2 pSya() + g^ySapOVSapy; reduced 

equations for g x = g 2 or g x = g 2 = # 3 ; 
II. [ 9l aSPyQ + giPSyaQ + mScfiO + a0Sj8y()]/So0y; 

reduced equation for gi = g 2 , or if a — 0; 
III. g + [(a/3 + cy)80yQ + bySya ()]/SaPy, reduced if 
a = = c, or a = = b = c. 

EXAMPLES 
(1). Let <pp=V-app, where SaP + 0. Take X = a, 
u = P, v = Vafi, then we find with little trouble 

mi = - Sap, m 2 = - a 2 /? 2 , ra 3 = a 2 p 2 SaP, 

and the characteristic equation of <p, 

(tp + Sap)(<p - Tap)(<p + Tap) = 0. 

Hence there are three invariant lines in general, and oper- 
ating on p by (<p + Safi)(<p — TaP), we find the invariant 
line corresponding, 

(<p + SaP)p = aSpp + pSap, 
(<p- TaP)(<p + Sap)p 

= a 2 pSPp + P 2 aSap - aTaPSPp - pTaPSap 
= - (TaSpp+ TpSap)(Ua+ UP)Tap. 

Hence the invariant line corresponding to the root TaP is 
Ua + Up. The other two are 

Ua - Up and UVap. 



THE LINEAR VECTOR FUNCTION 229 

(2). Let <pp = Vafip. 

(3). Let <pp = g 2 aS(3yp + frtfSyap + ySo&p) + hfiSaPp. 

(4). Let <pp= gp+ (fifi + ly)Sfap + r(3Syap. 

(5). Let <pp = Vep. 

6. Solution of cpp = a. It is obvious that when <p satis- 
fies the general equation 

<p 3 — mi<p 2 + m 2 (p — ra 3 = 0, ra 3 4= 0, 

then the vector 

m%<p~ l p = (w 2 — miv? + <^ 2 )p. 

For if we take the <p function of this vector, we have an 
identity for all values of p. Also this vector is unique, for 
if a vector a had to be added to the left side, or could be 
added to the left side, then it would have to satisfy the 
equation <pa = 0. But if ra 3 4= 0, there is no vector satis- 
fying this equation, for this equation would lead to a 
zero root for <p. Hence, if cpp = X, ra 3 p = m 2 X — mnp\ + <p 2 X, 
which solves the equation. 

If <p satisfies the general equation 

(pi — mnp 2 + m<2<p = 0, m% #= 0, 

then we have one and only one zero root of the latent equa- 
tion, and corresponding to it a unique vector for which 
<pa = 0, and if (pp = X, 

m 2 p = xa + [m\(p — (p 2 )p = xa + w&iX — <pX. 

If (p satisfies the cubic 

(p z — rriiv 2 =0, mi 4 0, 

the vacuity is two, and we have two cases according as 
there is not a reduced equation, or a reduced equation exists 



230 VECTOR CALCULUS 

of the form <p 2 — m\<p = 0. In either case the other root 
is mi. There is a corresponding invariant line X, and if the 
vector a is such that <pa = 0, then we have in the two cases 
a vector (3 such that respectively <p(3 = a, or <p(3 = 0. 
Hence, if <pp = 7, we must have in the two cases 

7 = x\ + yot, or 7 = x\. 

Otherwise the equation is impossible. Hence 

mip = x\ + za + yj3 = 7 + ua -f 2/0, 

where ^>/3 = a, <pa = 0, or where <pfi = = ^ck- 

If ^> satisfies the cubic 

and no reduced equation, there are three vectors (of which 
fi and 7 are not unique) such that <py = fi, <fP = a, <^a = 0, 
and then <pp = X, we must have X = xa + yft where p is 
any vector of the form 

p = za + iCjS + 1/7. 

If <p 2 = 0, and no lower degree vanishes, then 

<p(x(3 + 2/7) = <*j ^a = 0, and X = ua. 

If <p = 0, there is no solution except for <pp = 0, where p 
may be any vector. 

7. Zero Roots. It is evident that if one root is zero, 
then the region <p\ where X is any vector will give us the 
other roots. For instance let <pp = Vep. Then if /x = Veh, 

cpp, = Xe 2 — eSe\, <p 2 fi = e 2 /x, 

and the other two roots are ± V — 1 • Te. 

If two roots are zero, then <p 2 on any vector will give the 
invariant region of the other root. For instance, let 



THE LINEAR VECTOR FUNCTION 231 

<pp = aSfiyp, then aSfiyaSfiyp = <p 2 p. Hence cpa = aSapy 
gives the other root as Sapy and its invariant line a. 

In case a root is not zero, but is g\, if it is of multiplicity 
one, then <p — gi operating upon any vector will give the 
region of the other root, or roots. If it is of multiplicity 
two, then we use (<p — g{) 2 on any vector. 

8. Transverse. We define now a linear vector operator 
related to <p, and sometimes equal to <p, which we shall 
indicate by <p' ', and call the conjugate of <p, or transverse of 
<p, and define by the equation 

S\<pijl = Sn<p'\ for all X, /z. 

For example, if <pp = Vap(3, then S\<pp = S\ap(3 = SpfiXa, 
and <p' = VPQa = <p, if <pp = Vep, <p'p = — Vep; if 
<pp = aSfip, <p'p = (3Sap. 
If a is an invariant line of <p, (pa — ga, then for every /S 

8p<pa = gSaP = Sa<p'P, 
or 

Satf - g)P = 0, 

that is a is perpendicular to the region not annulled by 
<p[ — g, that is invariant for <p' — g. If we consider that 
from the definition we have equally 

S\(p 2 p, = Sfxcp' X, S\(p z iJL = Sup' X, 

it is clear that <p and <p' have the same characteristic equa- 
tion and the same general equation. They can differ only 
in their invariant regions if at all. If then the roots are all 
distinct, it is evident that the invariant line a of <p, is normal 
to the two invariant lines of <p' corresponding to the other 
two roots, hence each invariant line of <p is normal to the 
two of <p' corresponding to the other roots, and conversely. 
If now the characteristic equation is the general equation, 



232 VECTOR CALCULUS 

so that each function satisfies only the general equation, 
let there be two equal roots, g, whose shear region gives 

<pa= ga + ft <p@ = g(3, let <py = giy. 

Then 

&Vp = gSap + Sj3p, S/Vp = gSfip, Sycp'p = 0i#yp, 
Sapy<p'p = g(V(3ySap + VyaSfo) + F)M0P 

+ giVa(3Syp. 

Therefore corresponding to the root g\, <p' has the in- 
variant line Vafi, and to the root g, the invariant line V(3y. 
Further (p f converts Vya into gVya + Vfiy. 

Hence the invariant line of g\ for <p' is normal to the 
shear region of g, and the shear region of g for <p f is normal 
to the invariant line of g\ for <p, but the invariant line of 
g for ip' is normal further to the shear direction of g for <p, 
and the shear direction of <p' for g is normal to the invariant 
line of (p for g. 

In case there are three equal roots, and no reduced equa- 
tion, we have 

<pa = ga + ft <p/3 = gfi + 7, <£>7 = PY, 

so that 

&Vp = gSap + Sft>, Wp = gS(3p + S 7 p, 

#7<p'p = gSyp, 

Sapy<p'p = p^Sa/fy + VfhfSfo + F7CKS7P. 

Hence, the invariant line of <p' is Vfiy, its first shear line 
Vya, and second shear line Vafi. 

In case there is a reduced equation with two distinct 
roots, we have 

<p(xa -f- y&) = 5f(ara + yfi), <P7 = 0i7, 

Sa<p'p = gSap, S/Vp = gSfip, Sy<p'p = giSyp, 

Sa&y-<p'p = gVfiySap + gVyaSfip + giVa(3Syp, 



THE LINEAR VECTOR FUNCTION 233 

Hence, the invariant line of <p' corresponding to gi is normal 
to the invariant plane of g for <p, corresponding to g there 
is an invariant plane normal to the invariant line of gi for <p. 
Every line in the plane through the origin is invariant. 
In case the reduced equation has two equal roots, then 

<pa = ga + ft <pP = gP, <py = gy, 

Sa<p'p = gSap + S(3p, Sy<p'p = gSyp, S(3<p'p = gSfip, 

Sa(3y<p'p = gp + Sfa-iVfa), 

Corresponding to g, we have then two invariant lines: 
Va.fi, which is perpendicular to the shear plane of <p; V(3y, 
which is perpendicular to the non-shear region of g and to 
the shear direction of g; also the shear direction of <p' is 
Vfiy, so that the shear region of <p' is determined by Vya 
and Vfiy, and is therefore perpendicular to y. 
The three forms of <p' are 

I. <p' = [giVfySaO + toVyaSpQ + g 3 Va(3Sy01ISapy; 

II. <p' = faVpySaQ + giVyaSpQ + aVfaSPQ 

+ g 2 VaPSyQ]/Safrr, 

III. <p' m g + [aVfiySPQ + bVyaSyQ + cVpySyQ]/SaPy. 

We may summarize these results in the theorem : 
The invariant regions of ip' corresponding to the distinct 
roots are normal to the corresponding regions of the other 
roots for <p. In case there are repeated roots, if there is a 
plane every line of which through the origin is invariant, 
then every line of the corresponding plane will also be in- 
variant, but if there is a plane with an invariant line and 
a shear direction in it, the first invariant line of the con- 
jugate will be perpendicular to the shear direction and to 
the second invariant line of <p, and the shear direction of the 
conjugate will be perpendicular to the invariant lines of ip; 

16 



234 VECTOR CALCULUS 

while finally, if there is an invariant line, a first shear direc- 
tion, and a second shear direction, then the invariant line 
of the conjugate mil be perpendicular to the invariant line 
and the first shear direction of <p, the first shear direction 
will be perpendicular to the invariant line and the second 
shear direction of <p, and the second shear direction will be 
perpendicular to the two shear directions of <p. Let a, /3, y 
define the various directions a = V(3y/Sa(3y, /? = Vya/Sa/3y, 
y = VaP/Sa(3y, then we have 

<p = gioSoi + gtffS0 + gzySy) 
<p' = giaSa + g 2 @S(3 +. g 3 ySy J 



or 



or 



( gi aSa + gJISp + afiSa + g 2 ySy) 
\ gi aSa + gi (3S(3 + aaS(3 + g 2 ySy\ 



ig+(aJl+cy)Sa+byS(3 
\g+aaSp+ (b(3 + ca)Sy . 

9. Self Transverse. It is evident now that <p = <p' only 
when there are no shear regions, if we limit ourselves to 
real vectors, and further the invariant lines must be per- 
pendicular or if two are not perpendicular, then every 
vector in their plane must be an invariant, and even in this 
case the invariants may be taken perpendicular. Hence 
every real self-transverse linear vector operator may be 
reduced to the form 

<pp = — aSapgi — (3S(3pg 2 — ySypg 3 , 

where a /3 y form a trirectangular system, and where the 
roots g may be equal. 

Conversely, when <p — <p', the roots are real, provided 
that we have only real vectors in the system, for if a root 
has the form g + ih, where i — V — 1, then if the invariant 



THE LINEAR VECTOR FUNCTION 235 

line for this root be X + ip,, where X and p are real, we have 
<p(\ + in) = (g + ih)(k + ifi) = g\ — hp + i(h\ + gp) 

= <p\-\- iipjJL. 

Therefore 

<p\ = g\ — hp, <pp = hX + gfx, 
and 

$/*<pX = gS\p — hp 2 = S\<pp = AX 2 + <7$Xju. 

Thus we must have 

^X 2 + hf? = 0. 

It follows that h = 0. 

Of course the roots may be real without <p being self- 
transverse. 

An important theorem is that <p tp' and <p'<p are self- 
transverse. For 

Sp<p(p'(T = Sa<p<p'p, Sp<p'<p<r — S(T(p f (pp. 

EXERCISE 

Find expressions for <p<p' and <p'<p in terms of a, /3, 7, a, jS, 7. 

10. Chi of p. We define now two very important func- 
tions related to <p and always derivable from it. First 

X* = m i — <P> 
so that 

Sa(3y-x<pP — pSafi<py + pS(3y<pa + pSyapfi — (paSfiyp 

— cpfiSyap — <pySa(3p 
= VVaPV<pyp + • • • 
= aSp((3<py — y<p(3) + • • •. 

The other function is indicated by \j/^ or by x vv and defined 

4r p = m 2 — mi(p + <p 2 = ra 2 — <px v , 
Sapy-i/z^p = pSonpfiipy + • • • — <paSp((3(py — y<p(3) 
= aSp<p(3(py -\~ f3Sp<py<pa + ySp<pa<p(3. 



236 VECTOR CALCULUS 

We have at once from these formulae the following im- 
portant forms for FX/x, 

X„FX/x = [aSVlniVpvy - Vy<pp) • • .]/SaPy 

= [aS(V<p'\n - V\<p'n)V0y + • • -]/SaPy 
= *W + V\i/>% 

Whence we have also 

<pV\p = miV\ix — V\<p'n — Vp'XfjL, 
1^FX/x = [aSV\»V<pp<py H ]/Sapy 

= V<p'\<p'lL. 

Since it s evident that 

X+ = x/j and \p v , = #/, 
we have at once 

x\V\ii = V<p\» + V\<pfi 

^FX/x = V(p\<pji. 

The two expressions on the right are thus shown to be 
functions of FX/x. 

It is evident that as multipliers of p 

™<i = <P + X = f' + X'i 
^2 - *>X + lA = *>'x' + ^', 

m 3 = ^ = ^V- 

EXERCISES 

1. If <p = aiSPiQ + a 2 SM) + «*Sfo(), show that 

<p' = faScHQ + 2 Sa 2 () +01&I.O, 
X = 27/9i7ai(), 
* = - 2F/3 1/ 8 2 5Fa l a 2 (), 
mi = 2*Sau9i, ra 2 = — Z£Faia 2 F(8i/3 2 , ra 3 = — Saia 2 azS0ifi 2 3 , 

X ' = UFaiV/SiO, 
^ = - HVaiatSVPiPiQ. 

2. Show that the irrotational invariants of x and ^ are mi(x) = 2m h 
m 2 (x) = »ii 2 + m 2 , w 3 (x) = Wim 2 — m 3 ; rai(^) = ra 2 , m 2 (^) = raira 3 , 
Wj(^) = m 3 2 . 



THE LINEAR VECTOR FUNCTION 237 

3. For any linear vector function <p, and its powers <p 2 , <p 3 , • • • , we have 

mi(<f?) = Wi 2 — 2ra 2 , m 2 (<p 2 ) = m 2 2 — 2raira 3 , w 3 (^) = ra 3 2 . 
mi(<p 3 ) = mi 3 — 3wiW 2 + 3m 3 , m 2 (<p 3 ) = 3raira 2 ra 3 — m 2 3 — 3ra 3 2 , 

m 3 (<p 3 ) = m 3 3 . 
mi(^ 4 ) = mi 3 — 4mi 2 w 2 + 2w 2 2 + 4raim 3 
w 2 (<p 4 ) = w 2 4 — 4wiw 2 2 w 3 + 2wi 2 m 3 2 + 4ra 2 ra 3 2 , m 3 (^> 4 ) = ra 3 4 . 

4. Show that for the function <? + c, where c is a scalar multiplier, 

mi(<p + c) = wi(^) + 3c, m 2 ((p + c) = 0ts(?) + 2mi(<p)c -f 3c 2 , 
wi 3 (¥> + c ) = w s(«p) + cw 2 (<p) + c 2 mi(<p) + c 3 . 

5. Study functions of the form x\p + ?/x + 2. 

6. <p'V<p\<pfi = m 3 V\n; <p'(V\<pn — F/x^X) = m 2 F\M — V<p\<pn. 

7. ^(a^>) = aHiv)', tifilPi) = ^(<Pi)-^('Pi)' 

8. «A(a) = a 2 , ^[7a()J = - aSaQ, *(- 0Sa) = 0. 

+{— QxiSi - gsjSj - g 3 kSk) = - g 2 g 3 iSi — g 3 gijSj - gig 2 kSk. 

= - VptfiSVaiyi - VPtppSVata, - VptfiSVa&n. 

10. For any two operators <p, 9, 

mi(<pd) = mi(M, m. 2 (<pd) = m 2 (6<p), m 3 (<pd) = tr»»(0*). 
mi((p6) = mi(<p)mi(0) + ra 2 (<p) + ra 2 (0) — m 2 {6 + <p). 
m 2 (<pd) = m 2 {6)m 2 (<p) + m 3 {<p)-m v {d) + ra 3 (0)-rai(y?) 

- mtffto + *'(*)]. 
m 3 (<pd) = m 3 (<p)-m 3 {6). 
rrii(<p + 0) = mi{<p) + Wi(0). 

m 2 (v? + 0) = m 2 (v?) + m 2 (0) + mi(6)'ini(ip) - nii(<pO). 
mt(<p + 0) = m 3 (*>) + m 3 (0) + mi[*V(4) + 0V(*>)]. 

11. x can have the three forms : 

t (ff* + 9t)*Sa + (g 3 + 0i)0S0 + (oi + g 2 )ySj; 
II. fo + o 2 )a£5 + fo + fln)/aSg + 2^x7^7 + apSa; 
III. 2g - (a/3 + c 7 )>S£ - bySfT 

The operator x is the rotor dyadic of Jaumann. 

12. The forms of \f/ for the three types are 

I. g 2 g g aSa + g 3 gi&Sl3 + gig 2 ySy; 
II. gig 2 aSZ + g 2 gi0S8 + 0i 2 7#t" - agtfSZ; 
III. o 2 - [O03 + (ab - gc)y]Sa - bgySfi. 



238 VECTOR CALCULUS 

13. An operator called the deviator is defined by Schouten,* and is 
for the three forms as follows: 

I. (l9i - 9* - gs)aSZ + (Itfi - 0* - gi)0S8 + (lg* - g x - g%)ySy'; 
II. (- fci - <7*)(«S5 + fiSfi) + (§0» - 2^)7^ + apSZ; 
III. (o£ + Cy)Sa + bySd. 

It is V<p = <p — S<p, where S(<p) = \m\. 

14. Show that if F (X, M ) - - F (m, X) then 

F (X, M ) - C (X, m)Q.VX m , 
where C is symmetric in X, n and Q is a quaternion function of VX/i. 
11. We derive from <p and ^' the two functions 

That there is a vector e satisfying this last equation, and 
which is invariant, is easily shown. For if we form 

™>z(<P — <p')> we find that 

S(<p - <p')\{<p - <p')n(<p - (p')v 

= S(p\<piJL<pi> — 2$ (p\<p' )jl<p' v — ^LSipkcpyup'v—Sip'Xv'mp'v 
= S\iAi>(m 3 — ra 3 + Wi(^» — mi(^')). 

But it is easy to see that this expression vanishes identically, 
for the first two terms cancel, and if <p lt <p 2 are any two linear 
vector functions, we have 

= Siiv<p\kSiiv<p<Lh + SjjLvcpiiiSvXip^K + SiiV(pivS\mp2K 

+ Sl>\<Pi\SlJlV<p2lJL + Sv\<PillSv\(p2lJL + Sp\<PipS\/JL(P2H 
+ S\fJL<Pi\SlJLV<p2V + S\lJL<PilJ,ST<P2V + $XjU<pi J/jSAjU^ 

= S^knv - mi(<p2<pi) . 

Hence we may under mi permute cyclically the vector 
functions. Again after this has been done we may take 
the conjugate. Hence the expression above vanishes, and 
there is a zero root in all cases for <p — <p'. Further we 
may always write 

* Grundlagen der Vector- und Affinor-Analysis, p. G4. 



THE LINEAR VECTOR FUNCTION 239 

S\fXV<pp = (pXSfJLPp + • • * 

S\jJLl>'<p'p = VjivS\<p f p + • • • 
= VfivSipKp + 
Hence we have 

S\nv(<p - <p') P = V P V(Vfip)cp\ + .... 

From this we have 2eS\pv = V(p\Vpv + • • • for every 
noncoplanar X, p, v. 

The function <p is evidently self-transverse, and the 
conjugate of VeQ is — VeQ. It is easy to show that 

2<peS\pv = — V\V<pp(pv — • • •. 

The expressions Te, T<pe, and Sepe are scalar invariants 
of <p, and these three may be called the rotational invariants. 
In terms of them and the other three scalar invariants all 
scalar invariants of <p or <p' may be expressed. 

If there are three distinct roots, g\, g 2 , g 3 , and the corre- 
sponding invariant unit vectors are y h y 2 , 73, we may set 
these for X, p, v, and thus 

2e&7iy 2 Y3 = giVjiVy 2 y3 + g 2 Vy 2 Vy z yi + gzVy z Vy^y 2 
= (92 — g3)yiSy 2 y 3 + (g 3 - gi)y2Sy s y 2 

+ (gi — g2)yzSyiy 2 . 
2<peSyiy 2 y 3 = - g 2 g 3 Vy x Vy 2 y z — g 3 g\Vy 2 Vy z y x 

— gig2Vy 3 Vyiy 2 . 

In case two roots are equal and (pa = g x a + (3h 2 , 
<p(3 = #i/3, (py = g 2 y, we have 

2eSa(3y = (g 2 - gi)VyVafi + VQVPyh. 

In case three roots are equal, <pa = ga-\-h(3, <pfi = gr/5+ ly, 
<py = gy 

2eSa(3y = h(3V(3y + lyVya. 

It is evident, therefore, that if the roots are distinct and 



240 VECTOR CALCULUS 

the axes perpendicular two and two, that « = 0; if two 
roots are equal and the invariant line of the other root is 
perpendicular to the plane of the equal roots, then it is the 
direction of e; and if the three roots are equal, and if the 
invariant line is perpendicular to the two shear directions, 
then € is in the plane of the invariant line and the second 
shear. 

12. Vanishing Invariants. The vanishing of the scalar 
invariants of (p leads to some interesting theorems. 

If Wi = 0, there is an infinite set of trihedrals which are 
transformed by <p into trihedrals whose edges are in the 
faces of the original trihedral. If ^transforms any trihedral 
in this manner, mi = 0, and there is an infinite set of trihe- 
drals so transformed. 

We choose X, n, v for the edges of the vertices, and if <p\ 
is coplanar with /z, 7, <pix with v, X, and <pv with X, ju, the 
invariant mi = 0. If mi = 0, we choose X, ju, arbitrarily, 
and determine v from ScpXnv = = SXcpuv. Then also 
S\n<pp = 0. 

The invariant m 2 vanishes if <p transforms a trihedral 
into another whose faces pass through the edges of the first. 
The converse holds for any infinity of trihedrals. 

EXERCISES 

1. Show that if a, fi, 7 form a trirectangular system 

mi = — Sa<pa — S/3<pl3 — Sy<py 

and is invariant for all trirectangular systems, 

m 2 (<p<p') = T*<poc + TV/? + T*<py, 
TV(X) = S 2 \<pa -f £ 2 Xv/3 + S 2 \<py. 

2. Study the functions for the ellipsoid and the two hyperboloids 

- <p = a^aSa ± b~ 2 fiSl3 ± c^ySy. 
3 Study the functions 

ZmVaVQct, <P + VaVQa, a^VoapQ, 

r-VpVaQ, V -vVaQ.fi. 



THE LINEAR VECTOR FUNCTION 



241 



4. Show that 



V <pp = 2e — mi, 
\/Sp<pp = — 2 (pop, 



VAp = — 2<pt — m 2 , 
\7Vp<pp = 2Sep + Wip — 3<pp, 



wherein <p is a constant function. Hence (pop may always be repre- 
sented as a gradient of a scalar, Sep as a convergence of a vector, and 
m,\p — 3<pp (deviation) as a curl. We may consider also that Wi is a 
convergence and e is a curl, ra 2 a convergence and <pe a curl. 

5. An orthogonal function is defined to be one such that 

ip<p' = 1. 

Show that an orthogonal function can be reduced to the form 

ip = () cos - sin 0-70/3 = (lT cos 9)0800 = 0**l*Q0-*l* 

or — /3(0/" - )+ 1 ()/? -(fl ./ 7r)-1 which is a rotation about the axis /3 through 
the angle — 0, or such a rotation followed by reflection in the plane 
normal to /?. 

6. Study the operator <p 112 . 

7. Show that 



m 


■i(<po) = m h 


m 2 


(<Po) = 


mi + e 2 , m 3 (<p ) = m 3 ■ 


f 5-6V56. 


Hence if 
Tf 


Te 


= 0, 


m 2 (<p) = m 2 (<p ). 




u 

8. 


Show that 


Se<pt 


i = 0, 


m 3 (<p) = m 3 (<p ). 






mAVe{)] 


= o, 


m*[VeQ] = TV, m 3 [Ve()] = 


0. 


9. 


Show that 












e(x) = - 


■ 6, 


«(x) 


- " * e ' e( ^ _1) = " m 3 


ipe. 


10, 
11, 


, Show that 

. if = y./M), 


\p(<Po) 


= * + aSe(). 






rni(d) = 


2S/3e, 


m-i 


(0) = - S/3*>/3, m 3 (d) = 


o, 



" 2 (a 2 — aSa), 



12. If *> = F-a(), 

,p 2 » = 

^2n+l = a 2n7 a () > 

13. For any two operators <p, 0, 

2eM) = 2e(^ o 0o) + X(<p)e(6) + x{0)e(<p) + V-e(<p)e(6). 



242 VECTOR CALCULUS 

In particular 

14. An operator ^> is a similitude when for every unit vector a, 
T^a = c, a constant. 

Show that the necessary and sufficient condition is 

<p'<p = c 2 . 

Any linear transformation which preserves all angles is a similitude. 

15. If <p = aSi + 0Sj + ySk, then <p' = iSa +jSp + kSy, and 
^j^' = — ctSa — fiSfi — 7$7, 

mi(*V) = Pa + P/3 + 7*7, m,(^^') = PFa/S + 7*7/37 + T^a, 
mz(<p<p') = — S 2 a0y. 

13. Derivative Dyadic. There is a dyadic related to a 
variable vector field of great importance which we will 
study next. It is called the derivative dyadic, since it is 
somewhat of the nature of a derivative, as well as of the 
nature of a dyadic. This linear vector function for the 
field of a will be indicated by D a and defined by the equation 

D.= ~ SQV-<r. 

It is evident at once that if we operate upon dp, we arrive 
at da. This function is, therefore, the operator which en- 
ables us to convert the various infinitesimal displacements 
in the field into the corresponding infinitesimal changes 
in the field itself. 
The expression 

SdpDJp = Cdf, 

where C is a constant and dt a constant differential, repre- 
sents an infinitesimal quadric surface, the normals at the 
ends of the infinitesimal vectors dp being D a dp. 

Let us consider now the field of a, containing the con- 
gruence of vector lines of <r. Consider a small volume 
given by 8p at the point whose vector is p, and let us sup- 



THE LINEAR VECTOR FUNCTION 243 

pose it has been moved to a neighboring position given by 
the vector lines of the congruence, that is, p becomes 
p + adt. Then p + 8p becomes 

p+8p + dt(<r + DM, 

that is to say, dp has become 

(1 + Dadt)8p. 

Hence any area V8\p8 2 p becomes, to terms of the first 
order only, 

V8 lP 8 2 p + dt(V8 lP D,8 2 p + VDJ lP 8 2P ). 

The rate of change with regard to t of the vector area 
V8ip8 2 p is therefore 

X (D ff )V8 lP 8 2 p. 

Likewise, the infinitesimal volume S8ip8 2 p8 s p is trans- 
formed into the volume 

S8ip8 2 p8 3 p + dt{S8ip8 2 pD a 8 z p + S8ipD a 8 2 p8 3 p 

+ SD a 8ip8 2 p8 3 p). 

The rate of increase of the volume is, therefore, miS8ip8 2 p8 3 p. 
In other words if we displace any portion of the space of 
the medium so that its points travel infinitesimal distances 
along the lines of the congruence of a, by amounts propor- 
tional to the intensity of the field at the various points, then 
the change in any infinitesimal line in the portion of space 
moved is given by dtD ff 8p, the change in any infinitesimal 
area is given by x'(D a )dt- Area, and the change in an 
infinitesimal volume is midt times the volume. 

In case a defines a velocity field the changes mentioned 
will actually take place. We have here evidently a most 
important operator for the study of hydrodynamics. If 
adt is the field of an infinitesimal strain, then D a 8p is the 



244 VECTOR CALCULUS 

displacement of the point at dp. Evidently the operator 
plays an important part in the theory of strain, and con- 
sequently of stress. Further, (we shall not stop to prove 
the result as we do not develop it) for any vector a a 
function of p we have an expansion analogous to Taylor's 
theorem, in the series 

h 2 
<r(p + ha) = (r(po) + hD^ + -^ (- &*V)Z).a 

+ | (SaV) 2 D a a + •••• 

This formula is the basis of the study of the singularities 
of the congruence. For if cr(p ) = 0, then the formula will 
start with the second term, and the character of the con- 
gruence will depend upon the roots of D ff . In brief the 
results of the investigation of Poincare referred to above 
(p. 38) show that if none of the roots is zero, we have the 
cases : 

1. Roots real and same sign, the singularity is a node. 

2. Roots real but not all of the same sign, a faux. 

3. One real root of same sign as real part of other two, 
a focus. 

4. One real root of sign opposite the real part of others, 
a faux-focus. 

5. One real root, other two pure imaginaries, a center. 
If one or more roots vanish, we have special cases to con- 
sider. 

The invariants of D a are easily found, and are 

mi = — SV<r, e = ^Vxja, m 2 = — %SVViV2V<tkt 2 , 
D*e = iV-VViV2V<ri<T2, m 3 = |SViV2V3<W- 2 0- 3 . 

After differentiation, the subscripts are all removed. The 
related functions are 



THE LINEAR VECTOR FUNCTION 245 

BJ = - v&r(), X = - VVV*Q, %' = ~ V-VQV-*, 

$ = - jrviV2^K7 2 (), y = - i&oviVs-Wi*!. 

In a strain a the dilatation 's ra b the density of rotation 
(spin) is e, and in other cases we can interpret m\ and e in 
terms of the convergence and the curl of the field. In 
case a is a field of magnetic induction due to extraneous 
causes, and a is the unit normal of an infinitesimal circuit 
of electricity, then %'« is the negative of the force density 
per unit current on the circuit. In any case we might call 
— x'V8i P 8 2p the force density per unit circuit. Since x' 
is not usually self-transverse, the force on circuit a has a 
component in the direction jS different from the component 
in the direction a of the force on circuit ft. 

Recurring to Stokes' and Green's theorems we see that 

fdpa = ff - WVd lP d 2P '(T 

= 2ffS8 lP 8 2 pe -Sfx'V8 lP 8 2P . 

It is clear that the circulation in the field of a is always 
zero unless for some points inside the circuit e is not zero. 
The torque of the field on the circuit vanishes for any 
normal which is a zero axis of x'« From these it is clear that, 
if we have a linear function <pd P , in order that it be an exact 
differential da we must have the necessary and sufficient 
conditions 

VVvO m 0. 

For if tf<pd P = 0, then <pVUvV = for all Uv, whence 
the condition. The converse is easy. 

The invariant m 3 in the case of the points at which a = 
will be sometimes positive, sometimes negative. A theorem 
given originally by Kronecker enables us to find what the 
excess of the number of roots at which ra 3 is positive over 
the number of roots at which ra 3 is negative is.* We set 

* Picard, Traite d'Analyse, Vol. I, p. 139. 



246 VECTOR CALCULUS 

t = fa/To* and 7 = - J- ffSdvr] 

47T 

then the integral will vanish for any space containing no 
roots, and will be the excess in question for any other space. 
We could sometimes use this theorem to determine the 
number of singularities in a region of space and something 
about their character. It is evident that <SVr = 0. 

The operator (D c ) = \(D + DJ) is called the deforma- 
tion of the field, and the operator Ve() the rotation of the 
field. 

In case a is a unit vector everywhere, then DJa = 0, 
and since the transverse has a zero root, D a itself must have 
a zero root. There is one direction then for which D ff a = 0. 
The vector lines given by Vadp = are the isogons of the 
field. In case there are two zero roots the isogons are any 
lines on certain isogon surfaces. 

EXERCISES 

1. Study the fields given by 

a = — p, a = Up/p 2 , a = Vap, a = aSfip, a = Vap/p 3 . 

2. Show that if a is a function of p, 

a + da = — V o[Spo<r — %Spo<ppo] — \V Po^V <* 
= VVoihVvpo - Wpo<PPo] - lSV<r, 

where Vo operates only on p , and <p = — <rSS7 0- The first form 
expresses a + da as a gradient and a term dependent on the curl of a, 
the second as a curl and a term dependent on the convergence of a. 
po is an infinitesimal vector. 

3. If a = FVr, D a = ZV. 

14. Dyadic Field. If <p is a linear vector operator de- 
pendent upon p, we say that <p defines a dyadic field. For 
every point in space there will be a value of <p. Since there 
is always one root at least for <p which is real, with an in- 
variant line, there will be for every point in space a direction 



THE LINEAR VECTOR FUNCTION 247 

and a numerical value of the root which gives the real 
invariant direction and root. These will define a con- 
gruence of lines and a numerical value along the lines. 
In case the other axes are also real, and the roots are distinct 
or practically distinct, there will be two other related con- 
gruences. The study of the structure of a dyadic field 
from this point of view will not be entered into here, but 
it is evidently of considerable importance. 

EXERCISES 

1. If <p = uQ, then the gradient of the field is Vw. The vorticity 
of the field is VV <p() — VVuQ. The gradient in any case is v'V, 
a vector. 

2. If <p = VaQ, the gradient is — V\7<r, the vorticity is 

QSS7*+D V m - x {D,r). 

3. If <p = <tSt(), the gradient is aSVr — D T <r, the vorticity is 
WvStQ + V<rD y (). The gradient of the transverse field is tS\7<t 

- DaT, the vorticity VX/tSvQ + VtD<j{). 

4. If ip = VadQ, the gradient is - 70(V)<r + VadV, the vor- 
ticity is 

S\7<T-d() +S*V'-0'Q -*S$[(V i)Q +£<r0(). 

For the transverse field we have 

the gradient is — 0'FV'o - — 0VV<r, 
the vorticity 7v W«r() + W'dVa'Q. 

5. If <p = D,r the gradient of the field is — VV, the concentration of 
<r, and the vorticity is D vva • The gradient of the transverse field is 

— V^Vo", while the vorticity is zero. 

6. If <p = VV0(), the gradient is FV0V, where both V's act on 0, 
and the vorticity is V 2 0() - V&V0(). 

7. If <p = De(a), the gradient is — 7^, the vorticity is Dvve<r- 

8. If f> = to, the gradient is 2e(07V0). 

9. For any <p 

Vm 1 = < P V +2e (FWO ), 
Vm 2 = 2 € {<pW<p' + FW' I, 
Vw 3 = 2 e [V (Vi M *' 
-t W Vx]. 



248 VECTOR CALCULUS 

15. The Differentiator. We define the operator — SQ V 
as the differentiator, and indicate it by D. It may be used 
upon quaternions, vectors, scalars, or dyadics. 

As examples we have, D being the transverse 

B v „ = VaD r () - VtD Q, D Sar - SQD.r + S()D T a, 
D Vaa = - VaD.Q, D mi M = mriDJ, 

D eM = e(DJ, D v = -S()V •*>(). 

16. Change of Variable. Let F be a function of p, and 
p a function of three parameters u, v, w. Let 

A = ad/du + f3d/dv + yd/dw, 

where a, /3, y form a right-handed system of unit vectors. 
Then we have the following formulae to pass from expres- 
sions in terms of p to differential expressions in terms of the 
parameters. 

AF = - AiS Pl VF t 
FA' A" = |FAi'A 2 "£Fpip 2 Fv'V", 
SA'A"A'" - - i<SAi'A 2 "A8 , "iSpiP2PsSV' V'V". 

As instances 

- SVv= A'VV'V, 
VA<r= VV"T(r"A'. 

Notations 
Dyadic products 
4>(a), <f>'(a), <f)Va( ), Va(f>( ), Hamilton, Tait, Joly, Shaw. 
<l>'a f a-4>, <j) X a, aX <j>, Gibbs, Wilson, Jaumann, Jung. 

Reciprocal dyadic 
4>~ l , Hamilton, Tait, Joly, Gibbs, Wilson, Burali-Forti, 

Marcolongo, Shaw. 
q~ l , Timerding. 
I6I" 1 , filie. 



THE LINEAR VECTOR FUNCTION 249 

The adjunct dyadic 
\j/ = m(f)'~ l , Hamilton, Tait, Joly, Shaw. 
WO2, Gibbs, Wilson, Macfarlane. 
R{a), Burali-Forti, Marcolongo. 
x((f>, (f>), Shaw. 
D4>~ 1 , Jaumann, Jung. 

The transverse or conjugate dyadic 
<f>', Hamilton, Tait, Joly. 
0, Taber, Shaw. 

<f> c , Gibbs, Wilson, Jaumann, Jung, Macfarlane. 
K(ct), Burali-Forti, Marcolongo. 
\b / , Elie. 

The planar dyadic 
X = Wi — (f> r , Hamilton, Tait, Joly. 
4>J — <f> c , Gibbs, Wilson. 
— </>/, Jaumann, Jung. 
CK(a), Burali-Forti, Marcolongo. 
x(0), Shaw. 

Self-transverse or symmetric part of dyadic 
<f>o t Hamilton, Tait, Shaw. 
$, Joly. 

<f> f , Gibbs, Wilson. 
[</>], Jaumann, Jung. 
D(a), Burali-Forti, Marcolongo. 
\ b /, Elie. 
\ b° / , Elie. In this case expressed in terms of the axes. 

Skew part of dyadic 
\{4> — </>') = V-e( ), Hamilton, Tait, Joly, Shaw. 
</>", Gibbs, Wilson. 
II, Jaumann, Jung. 
Va A , Burali-Forti, Marcolongo. 

17 i 



250 VECTOR CALCULUS 

\ b / , £lie. 

Sin <f>, Macfarlane. 

Mixed functions of dyadic 
X«>, 0), Shaw. 
\<f>l 0, Gibbs, Wilson. 
R{(f>, 0), Burali-Forti, Marcolongo. 

Vector of dyadic 
e, Hamilton, Tait, Joly. 
<£ x , Gibbs, Wilson. 
(f> r 8 , — <}>/, Jaumann, Jung. 
Va, Burali-Forti, Marcolongo. 
E, Carvallo. 
R = Te, filie. 
c(<£), Shaw. 

Negative vector of adjunct dyadic 
<f>e, Hamilton, Tait, Joly. 
0-0 x , Gibbs, Wilson. 
<t>-<f> r 8 , Jaumann, Jung. 
olVol, Burali-Forti, Marcolongo. 
«x(</>> <f>)> Shaw. 

Square of pure strain factor of dyadic 
4><f>', Hamilton, Tait, Joly. 
</></> c , Gibbs, Wilson. 
{(f)} 2 , Jaumann, Jung. 
aKa, Burali-Forti, Marcolongo. 
[6], filie. 
</></>', Shaw. 

Dyadic function of negative vector of adjunct 
<f> 2 e, Hamilton, Tait, Joly, Shaw. 
<f> 2 -4> x , Wilson, Gibbs. 



THE LINEAR VECTOR FUNCTION 251 

2 -0/, Jaumann, Jung. 

a 2 Va, Burali-Forti, Marcolongo. 

K 2 , Elie. 

Scalar invariants of dyadic. Coefficients of characteristic 

equation 
m" ', ra', m, Hamilton, Tait, Joly, Carvallo. 
1%, h, h, Burali-Forti, Marcolongo, Elie. 
F, G, H, Timerding. 
S , (</>2) s , 03, Gibbs, Wilson, 
mi, ra 2 , ra 3 , Shaw. 

fc, ] 

4> 8 *, >■ • • • 03, Jaumann, Jung. 

- w, J 

cos </>••• 03, Macfarlane. 

(Mer scalar invariants 
™>i(<f>o 2 ), mi(00'), 2(rai 2 — m 2 ), rai(00')> 

wi[x(0, *)> 0L Shaw. 
[0 8 ] 2 «, {0j s 2 > [01/, •'* -j Jaumann, Jung. 
• • •, • • •, • • ., : 0, 0* : ft Gibbs, Wilson. 
Elie uses ifi for $e0e. 

Notations for Derivatives of Dyadic 
In these V operates on unless the subscript n indicates 
otherwise. 

Gradient of dyadic 
V0, Tait, Joly, Shaw. 

Dyadic of gradient. Specific force of field 
0V, Tait, Joly, Shaw, 
grad a, Burali-Forti, Marcolongo. 

-3 — , Fischer. 
dr 



252 VECTOR CALCULUS 

Transverse dyadic of gradient 
0'V, Tait, Joly. 
grad Ka, Burali-Forti, Marcolongo. 

—r^-y Fischer. 

V -<t>, Jaumann, Jung. 

Divergence of dyadic 

- SV<f>( ), Tait, Joly, Shaw. 

X grad Ka, Burali-Forti, Marcolongo. 

Vortex of dyadic 
VV4>( ), Tait, Joly, Shaw. 
Rot a, Burali-Forti. 

V X 0, Jaumann, Jung. 

Directional derivatives of dyadic 

- S( ) V • 0. Sa' 1 V ■ <l>a. ScT 1 V -<t>Va(), Tait, Joly, Shaw. 
S(a, ( )), Burali-Forti. 

P , IX*, F i sch e r . 
da da 

Burali-Forti, Marcolongo. 



(»<>)<»• 



Gradient of bilinear function 
ju„(Vn, «), Tait, Joly, Shaw. 
<£(/z)a, Burali-Forti. 

Bilinear gradient function 
ju(Vn, u n ), Tait, Joly, Shaw. 
\//(n, u), Burali-Forti. 

Planar derivative of dyadic 
<f> n VVn( ), Tait, Joly, Shaw. 

X-^> Fischer, 



CHAPTER X 
DEFORMABLE BODIES 

Strain 

1. When a body has its points displaced so that if the 
vector to a point P is p, we must express the vector to the 
new position of P, say P', by some function of p, cpp, 
then we say that the body has been strained. We do not 
at first need to consider the path of transition of P to P'. 
If cp is a linear vector function, then we say that the strain 
is a linear homogeneous strain. We have to put a few 
restrictions upon the generality of <p, since not every linear 
vector function can represent a strain. In the first place 
we notice that solid angles must not be turned into their 
symmetric angles, so that SipKcpyupvlSKp.v must be positive, 
that is, ra 3 is positive. Hence (p must have either one or 
three positive real roots. The corresponding invariant lines 
are, therefore, not reversed in direction. 

2. When <p is self-conjugate there are three real roots 
and three directions which form a trirectangular system. 
The strain in this case is called a pure strain. Any linear 
vector function can be written in the form 

#rr V.{**9."f I 0« = p -1 V(<pV)()-p, 

where 

q-i()q = (wT'V 

The function <p<p' is self -conjugate and, therefore, has three 
real roots and its invariant lines perpendicular. If we set 
7r = V ((p<p') y then 7r 2 = ipip'. Let the cubic in <p<p' be 
G 3 - M X G 2 + M 2 G - M 3 = 0. Then from the values 
given in Chapter IX, p. 237, for the coefficients of <p 2 

253 



254 VECTOR CALCULUS 

in terms of those of <p we have (the coefficients of the cubic 
in w being p u p 2 , p 3 ) 

Mi = pi* - 2p 2f M 2 = p 2 2 — 2pm, M 3 = p z 2 , 
whence we have 

P! 4 - 2(Mi + 8M 3 )pi 2 - \m 2 M zVl + MS - 4M 2 2 M 3 = 0. 

Thence we have pi, p 2 , and p 3 . 

Now if the invariant lines of <p<p' are the trirectangular 
unit vectors a, 0, 7, we may collect the terms of <p in the form 

<P = aaSa'Q + bpSP'Q + cySy'Q, 

where a, b, c are the roots of V <p<p' = w and a! , fi', 7' are 
to be determined. Hence <p' = aa'SaQ + • • • and 

- tp'ip = tfa'Sa'Q + VP'Sp'Q + WO- 

But also 

^' = _ otefo _ fc0S0 - c 2 7#7, 

since a, /?, 7 are axes of <p(p', and a 2 , b 2 , c 2 are roots. Now 
we have 

<p'a = — act', <p'/3 = — b(3', <p'y = — cy', 

hence 

<p(p r a = a 2 a = — a 2 otSa'ct' — ab(3Sa'(3' — acySot'y'. 

Thus we have a:' 2 = — 1, Set' (3' = = Sa'y', and similar 
equations, so that a', (3', y' are unit vectors forming a tri- 
rectangular system, and indeed are the invariant lines of 
<p'<p. We may now write at once 

7r = — aaSct — b/3S(3 — cySy, 
q ~\) q = - aSa' - fiSfi* - ySy'. 

This operator obviously rotates the system a', (3', y' into 



DEFORM ABLE BODIES 255 

the system a, (3, y, as a rigid body. That the function is 
orthogonal is obvious at a glance, since if we multiply it 
by its conjugate we have for the product 

- aSa - PSP - ySy = 1(). 

Reducing it to the standard form of example five, Chapter 
IX, p. 236, we find that the axis is UV(aa' + (3(3' + 77') and 
the sine of the angle of rotation \TV{aa! + $8' + 77') • 

EXAMPLES 

(1). Let <p = VeQ- Then 

<p' = - VeQ, <P<P' = ~ VeVeQ = eSe() - e 2 . 

The axes are e for the root 0, and any two vectors a, j8 
perpendicular to e, and these must be taken so that a(3 = Ue, 
the roots that are equal being T 2 e. We may therefore write 

<p = TeaS(3 - Te(3Sa = V-eQ, 

which was obvious anyhow. Hence we have for q~ l Qq the 
operator 

aSp-0Sa= V(VaP)Q, 

and this is a rotation of 90° about Va(3 = Ve of 90°. The 
effect of 

7T = Te(- aSa - (3S(3) 

is to give the projection of the rotated vector on the plane 
perpendicular to e, times Te. That is, finally, VeQ rotates 
p about € as an axis through 90° and annuls the component 
of the new vector which is parallel to e. 

(2). Consider the operator g — aS(3Q where a, jS are any 
vectors. It is to be noticed that we must select of all the 
square roots of (p<p f that one which has its roots all positive. 
It is obvious that j) = q. 



256 VECTOR CALCULUS 

3. The strain converts the sphere Tp= r into the ellip- 
soid 7V -1 p = r, or 

WW = - r\ 

This is called the strain ellipsoid. Its axes are in the direc- 
tions of the perpendicular system of (p<p' — tt 2 . The ellip- 
soid Sp<p'<pp = — r 2 is converted into the sphere Tp = r. 
This is the reciprocal strain ellipsoid. Its axes are in the 
directions of the principal axes of the strain. The exten- 
sions of lines drawn in these directions in the state before 
the strain are stationary, and one of them is thus the maxi- 
mum, one the minimum extension. 

4. A shear is represented by 

<PP — P ~ fiSap, 

where Sa/3 = 0. The displacement is parallel to the vector 
/3 and proportional to its distance from the plane Sap = 0. 
There is no change in volume since ms = 1. 

If there is a uniform dilatation and a shear the function is 

<pp = gp- fiSap. 

The change in volume is now g 3 . The equation is easily 
seen to be 

(<P - 9? = 0. 

This is the necessary and sufficient condition of a dilatation 
and a shear, but this equation alone will not give the axes 
and the shear plane, of course. 

5. The function <pp = gqpcT 1 ~ qfiq~ l Sap is a form into 
which the most general strain can be put which is due to 
shifting in a fixed direction, U(5, planes parallel to the fixed 
plane Sap = by an amount proportional to the perpen- 
dicular distance from the fixed plane, then altering all 
lines in the ratio g } and superposing a rotation. This is 



DEFORMABLE BODIES 257 

any strain. We simply have to put <p'<p into the form 

<p'<p = b 2 + X£ju + ftSk, 
where 

S\fx = i(a 2 + c 2 - 2b 2 ), T\n = K« 2 - c 2 ), 
and then we take 

g =b, a - - X, bp = n- IXrK* ~ c ) 2 - 

The rotation is determined as before. 

6. All the lines in the original body that are lengthened 
in the same ratio, say g, are parallel to the edges of the cone 
TcpUp = g or SUp(<p f <p — g 2 )Up = 0, or in terms of X, /z, 
2SX UpSfx Up = b 2 - g 2 , sin u-smv= (b 2 - g 2 )/(a 2 - c 2 ) , 
where u and v are the angles the line makes with the cyclic 
planes of the cone Scpp<pp = — T 2 p. 

7. The displacement of the extremity of p is 

5 = a — p = O — l)p, 

which can be resolved along p and perpendicular to p into 
the parts 

p(/Sp~Vp — 1) + pVp~ l <pp. 

The coefficient of p in the first term is called the elongation. 
It is numerically the reciprocal of the square of the radius 
of the elongation quadric: 

Sp(<p — l)p = — 1, 

the radius being parallel to p. 

The other component may be written Vep + Vcpopp~ l - p, 
where e is the invariant vector of <p, the spin-vector. 

8. If now the strain is not homogeneous, we must con- 
sider it in its infinitesimal character. In this case we have 
again the formula da = — SdpV -tr = cpdp, where a is now 
the displacement of P, whose vector is p, and a + da that of 



258 VECTOR CALCULUS 

p -f- dp, provided that we can neglect terms of the second 
order. If these have to be considered, 

da = - SdpV a + i(SdpV) VV 
= (pdp — %SdpV - (pdp. 

We may analyze the strain in the case of first order into 

<P = (fo + VeQ. 

Since now € = \V\7<r, if e = 0, it follows that a = VP 
and there is a displacement potential and 

p '« - VSVP(). 

The strain is in this case a pure strain. If e is not zero, 
there is rotation, about e as an axis, of amount Te. In any 
case the function <p determines the changes of length of all 
lines in the body, the extension e of the short line in the 
direction Up being 

— SUppoUp. 

The six coefficients of <p , of form — Sa<po(3, where a, ft 
are any two of the three trirectangular vectors a, ft 7, 
are called the components of strain. Three are extensions 
and three are shears, an unsymmetrical division. 

9. In the case of small strains the volume increase is 
— S\7<7, and this is called the cubic dilatation. If it 
vanishes, the strain takes place with no change of volume, 
that is, with no change of density. A strain of this char- 
acter is called a transversal strain. There is a vector 
potential from which a can be derived by the formula 

a = VVt, SVt = 0. 

There is no scalar potential since we do not generally have 
also VVo- = 0. Indeed we have 

2 e = VV<r = WVVr = V 2 r - VSVr = V 2 r. 



DEFORMABLE BODIES 259 

This would give us the integral 

\t = \irfffejr-dv. 

The integration is over the entire body. 

This strain is called transverse because in case we have a 
a function of a single projection of p, on a given line, say a, 
so that 

a = af v x + /3f 2 -x + yf 3 -x, 
SVo- = — /i = 0, fi = constant, 

and all points are moved in this direction like those of a 
rigid body. We may therefore take the constant equal to 
zero, and /i = 0, so that 

Saa = 0. 

Hence every displacement is perpendicular to the line a. 

10. When V\/a = 0, we call the strain longitudinal; for, 
giving <j the same expression as in § 9, we see that we have 

Wa =0 = 7/2' - fifs', and / 2 = = / 3 , 
Vaa = 0. 

Hence we have all the strain parallel to a. 

11. In case the cubical dilatation iSVo" = 0, the strain 
is purely of a shearing character, and if the curl VVv = 0, 
the strain is purely of a dilatational character. Since any 
vector a can be separated into a solenoidaJ and a lamellar 
part in an infinity of ways, it is always possible to separate 
the strain into two parts, one of dilatation only, the other 
of shear only. 

If we write a = VP + V\/t, then we can find P and r 
in one way from the integrals 

P = lir.ffSS<T'VTp-W, 
r = - \TT'fffVa'VTp- l -dv f , p = p' - Pc . 



260 VECTOR CALCULUS 

The integrations extend throughout the body displaced. 
This method of resolution is not always successful, and 
other formulae must be used. (Duhem, Jour, des Math., 
1900.) 

12. The components are not functionally independent, 
but are subject to a set of relations due to Saint Venant. 
These relations are obvious in the quaternion form, equiva- 
lent to six scalar equations. The equation is 

V-V<PoVV() = 0, if <p=SQV-<r, 

where both V's operate on <p . The equation is, further- 
more, the necessary and sufficient condition that any linear 
vector function <p can represent a strain. The problem of 
finding the vector a when <p is a given linear and vector 
function of p consists in inverting the equation 

<p = — S() V -cr. (Kirchkoff, Mechanik, Vorlesung 27.) 

It is evident that if we operate upon dp, we have 

<pdp = do. 

Hence the problem reduces to the integration of a set of 
differential equations of the ordinary type. 

EXAMPLES 

(1). If (p = VeQ, we have or = Vep. Prove Saint Venant's 
equations. 

(2). If <p~ p- l V{)p-\ then a = Up. Prove Saint Ven- 
ant's equations. 

13. In general when we do not have small strains, we 
must modify the. preceding theory somewhat. The dis- 
placement will change the differential element dp into 

dpi = dp — SdpV-<r. 

The strain is characterized when we know the ratio of the 
two differential elements and this we may find by squaring 



DEFORMABLE BODIES 261 

so as to arrive at the tensor 

(dpi)* = Sdp[l - 2vSa + V'S(r'(T"SV"]dp- 

The function in the brackets is the general strain function, 
which we will represent by <£. It is easily clear that if 
<p = — SQV'<r then 

* = (1 + <p)(l + <p') = (1 + *>)(1 + <p)'. 

Of course $ is self-conjugate. Its components Sa&fi are 
also called components of strain. If <p is infinitesimal, we 
may substitute (1 + 2<po) for <£. 

The cubical dilatation is now found by subtracting 1 from 

SdipidhpidtPi/Sdipdtpdtp = m 3 (l + <p) = 1 + A. 

Evidently (1 + A) 2 = m 3 ($). The alteration in the 
angle of two elements is found from 

- suq. + <p)\u(i<p)y. 

If angles are not altered between the infinitesimal elements, 
the transformation is conformal, or isogonal. In such case 

Eti&k' = s 2 \ys\$\s\'$y. 

For example, if <p = VaQ, 

sua + <p)\Q. + <p)v = sxx', 

when Sa\ = = Sa\'. 

14. This part of the subject leads us into the theory of 
infinitesimal transformations, and is too extensive to be 
treated here. 

On Discontinuities 

15. If the function <j is continuous throughout a body, 
it may happen that its convergence or its curl may be dis- 
continuous. The consideration of such discontinuities is 



262 VECTOR CALCULUS 

usually given at length in a discussion of the potential 
functions. Here we need only the elements of the theory. 
We make use of the following general theorem from analysis. 

Lemma. If a function is continuous on one side of a sur- 
face for all points not actually on the surface in question, and 
if, as we approach the surface by each and every path leading 
up to a point P, the gradient of the function, or its directional 
derivatives approach one and the same limit for all the paths; 
then the differential of this function along a path lying on the 
surface is also given by the usual formula, 

— SdpV -q = dq, dp being on the surface. 

[Hadamard, Lecons sur la propagation des ondes, etc., 
p. 84, Painleve, Ann. Ecole Normale, 1887, Part 1, ch. 2, 
no. 2.] 

In the case of a vector a which has the same value on 
each side of a surface, which is the value on the surface, 
and is the limiting value as the surface is approached, at 
all points of the surface, we have on one side of the surface 

da — — Sdp\7 •& = <pidp. 

On the opposite side 

da = — Sdp\7 -<r = <p2dp. 

If now these two do not agree, but there is a discontinuity 
in <p, so that <p 2 — tp\ is finite as the two paths are made 
to approach the surface, then designating the fluctuation or 
saltus of a function by the notation [], we have in the limit 

[da] = (<p 2 — <Pi)dp = [<p]dp. 

But since a does not vary abruptly, [da] along the surface 
is zero, hence for dp on the surface 

[<p]dp = 0, 



DEFORMABLE BODIES 263 

and therefore 

M = — vSv> 

where v is the unit normal, \x a given vector. That is to 
say, we have for the transition of the surface 

[S()V-a] = »Sv. 
Whence 

[SVcr] = Spix, 
[W<t] = Vvix. 

These are conditions of compatibility of the surface of dis- 
continuities and the discontinuity; or identical conditions, 
under which the discontinuities can actually have the sur- 
face for their distribution. 

16. If *S/x^ = 0, then [S Vo"] = 0, and the cubic dilatation 
is continuous. 

Since Svvjjl = = Sv[V\7<t] = [SpV<t], the normal com- 
ponent of the curl of a is continuous, and the discontinuity 
is confined to the tangential component. Likewise 

Sfivn = = [S/xVo-], 

and the component along ijl is continuous. Hence V\7(r 
can be discontinuous only normal to the plane of /jl, v. 

17. In case a itself is discontinuous, the normal com- 
ponent of a as it passes the surface of discontinuity cannot 
be discontinuous without tearing the surface in two. Hence 
the discontinuity is purely tangential. It can be related 
to the curl of a as follows. . . 

Consider a line on the surface, of infinitesimal length, and 
an infinitesimal rectangle normal to the surface, and let 
the value of a at the two upper points differ only infinites- 
imally, as likewise at the two lower points, but the differ- 
ence at the two right hand points or at the two left hand 



264 VECTOR CALCULUS . 

points be finite, so that a has a discontinuity in going 
through the surface equal to [a]. Then 

fSbpa = ffSK(AWa) 

around the rectangle, when k is normal to the rectangle. 
But the four parts on the left for the four sides give simply 

Sid*}, 

where 8p is a horizontal side and equal to V-wTSp. Hence 
we have for every k tangential to the surface 

SkV v[a] - Sk Urn (AW<r)IT8p. 

Dropping all infinitesimals, we have 

Vv[(t] = Lim AVVcr/Tdp. 

Tangential discontinuities may therefore be considered 
to be representable by a limiting value of the curl multi- 
plied by an infinitesimal area, as if the surface of discon- 
tinuity were the locus of the axial lines of an infinity of small 
rotations which enable one space to roll upon the other. 
The expression \[<j] is the strength of this sheet. 

A strain is not irrotational unless such surfaces of dis- 
continuity are absent. But we have shown above that a 
continuous strain may imply certain surfaces of discon- 
tinuity in its derivatives of some order. If V\7cr = 0, 
everywhere, then Vv[u] = 0, and such discontinuity as 
exists is parallel to v. 

The derivation above applies to any case, and we may 
say that if a field is irrotational, any discontinuity it pos- 
sesses must be normal to the surface of discontinuity. 

Integrating in the same way over the surface of a small 
box, we would have 

ffSv[<r]ds = SV<T'V, 



DEFORMABLE BODIES 265 

where v is the infinitesimal volume. But this gives 

Sv[a] = vSV (r/surface. 

If then $Vo" = everywhere, the discontinuity of a is 
normal to the normal, that is, it is purely tangential. These 
theorems will be useful in the study of electro-dynamics. 

Kinematics of Displacements 
18. In the case of a continuous displacement which takes 
place in time we have as the vector a the velocity of a 
moving particle, and if p is the vector from a fixed point 
to the particle, then dp/dt = a. It is necessary to distin- 
guish between the velocity of the particle and the local 
velocity of the stream of particles as they pass a given fixed 
point in the absolute space which is supposed to be sta- 
tionary. The latter is designated by d/dt. Thus dcr/dt is 
the local rate of change of the velocity at a certain point. 
While da/dt is the rate of change of the velocity as we follow 
the particle. It is easy to see that for any quaternion q 
the actual time rate of change is 

dq/dt = dq/dt — SaV -q. 

We have thus the acceleration 

da/dt = da/dt - SaV-cr = (d/dt + <p)a. 

If the infinitesimal vector dp is considered to be displaced, 
we have 

bdp/dt = - S5pV'(r. 

Since the rotation is \V\7a dt, the angular velocity of turn 
of the particle to which dp is attached is |FVo". This is 
the vortex velocity. Likewise the velocity of cubic dilata- 
tion is — S\/a. 

The rate of change of an infinitesimal volume dv as it 
18 



266 VECTOR CALCULUS 

moves along is 

— SV<T'dv. 

The equation of continuity is d(cdv) = 0, where c is the 
density, or 

dc/dt + c{- SV<r) = 0. 

That is, we have for a medium of constant mass 

dc/dt = cSVv- 

That is, the density at a moving point has a rate of change 
per second equal to the density times the convergence of 
the velocity. 

It may also be written easily 

dc/dt = SVW. 

This means that at a fixed point the velocity of increase in 
density is equal to the convergence of the momentum per 
cubic centimeter. 

19. When FVo" = 0, the motion is irrotational, or dila- 
tational, and we may put a = VP, where now P is a veloc- 
ity-potential, which may be monodromic or polydromic. 
When SVcr = 0, the motion is solenoidal or circuital, and 
we may write a = VVr where &Vr = 0. r is the vector 
potential of velocity. The lines e = \V\7<r become in this 
case the concentration of Jr. The lines of a are the vortex 
lines of r, and the lines of e are the vortex lines of a. 

20. If a is continuous, and the equation of a surface of 
discontinuity of the gradient dyadic of a and of a' is / = 0, 
where now a is a displacement and a' is da/dt the velocity, 
we have certain conditions of kinematic compatibility. 
These were given by Christoffel in 1877-8 and are found as 
follows. We have 

M = o, [_0ov«W<-jtfi> 



DEFORMABLE BODIES 267 

in the case in which the time t is not involved; and for a 
moving surface in which / is a function of t as well as of p, 
we would have 

[-SOV-<r]=-»SUvfO, 

[" S Tt V<T ] = " mS i Uvf= M f /*V/=M=-Gm. 

This gives us the discontinuity in the time rate of change of 
the displacement of a point as it passes from one side to the 
other of the moving surface. The equation of the surface 
as it moves being /(/>, t), we have in the normal direction 

- SdpV-f+dtf = 0, 

that is, since dp is now Uvfdn, dn/dt — — f'/T\/f = G, 
where / ' is the derivative of / as to t alone. In words, 
at any point on the instantaneous position of the moving 
surface the rate of outward motion of the point of the 
surface coinciding with the fixed point in space is 
G = —f'jTS/f. The moving surface of discontinuity is 
called a wave and G the rate of propagation of the wave at 
the given point. We may now read the condition of com- 
patibility above in these words: the abrupt change in the 
displacement velocity is given by a definite vector p. at 
each point multiplied by the negative rate of propagation 
of the wave of displacement, that is, if G is the rate of 
propagation, 

[o-'j = - Gp, and [SVff] = - SpVvf = - S/iv. 

21. The preceding theorem becomes general for discon- 
tinuities of any order in the following way. Let the func- 
tion a and all its derivatives be continuous down to the 
(n — l)th, then we can write 

[SQiV'SQzV — S0*-iV-*]«0, 



268 VECTOR CALCULUS 

whence, differentiating along the surface of discontinuity 
as before, we find in precisely the same manner 

• [S()iV • • • SO. V •*) = nSOiUvfSihUvf • • • SQnUvf, 

since at a given point on the fixed surface V/ is constant. 
And if we insert dp/dt in m parentheses (m < ri), we 
shall have, since the surface is moving, 

= - »G»SQiUvf • • • S0n-mUvf(-l) m . 

In particular for m — 2 = n, we have 

W) = mG 2 , 

which is the discontinuity in the acceleration of the dis- 
placement. 

If m = 1, n = 2, 

[SOW] = - nGSQUVf. 

From this we derive easily 

[SVff'l = - GS»Uvf= - GSfxp. 
[W<r'] = - GVfxUvf= ~ GVfip. 

22. The nth. derivatives of Saa are 

[S()iV • • • SQnV-Saa] = SQiUvf • • ■ SQ n UVfSap. 

If then we hold the surface fixed and consider a certain 
point, the discontinuity in the nth derivative of the ratio 
of two values of the infinitesimal volume which has two 
perpendicular directions on the surface and the third along . 
the normal will be given by the formula 

SQiUvf ■ ■ ■ SOnllVfSnUvf. 
In case we have a material substance that has mass and 



DEFORMABLE BODIES 269 

density and of which the mass remains fixed, we have 

c/cq = volo/voi, 
log c — log Co = log v — log V, 
V log c = — V log v/v = — Vo/v-V(clvo). 

Therefore from the formula above we have since v /v = 1 
in the limit 

[SOiV • • • S()nV log c] = SQiUVf • • • SQnUvfSfjiUvf. 

In particular for the case of discontinuities of order two- 
we have 

[Vlogc]= UvfSfiUvf. 

23. These theorems may be extended to the case in which 
the medium is in motion as well as the wave of discontinuity. 

Stress 

24. In any body the stress at a given point is given as a 
tension or a pressure which is exerted from some source 
across an infinitesimal area situated at the point. The 
stress real y consists of two opposing actions, being taken 
as positive if a tension, negative if a pressure. It is as- 
sumed that the stress taken all over the surface of an 
infinitesimal closed solid in the body will be a system of 
forces in equilibrium, to terms of the first order. This is 
equivalent to assuming that the stress on any infinitesimal 
portion of the surface is a linear function of the normal, 
that is 

6 = ZVv. 

25. We have therefore for any infinitesimal portion of 
space inside the body 

ffQdA = ffZdv = 0. 

But by Green's theorem this is equal to the integral through 



270 VECTOR CALCULUS 

the infinitesimal space J J VHV = 0. Hence SV = 0. 
In this equation S is a function of p, and V differentiates S. 

26. In case the portion of space integrated over or 
through is not infinitesimal, this equation (in which S is 
no longer a constant function) remains true if there is 
equilibrium; and if there are external forces that produce 
equilibrium, say £ per unit volume, then the density being 
c, we have 

SV + c£ = 
for every point. 

In case there is a small motion, we have 

EV + c£ = co". 

27. Returning to the infinitesimal space considered, we 
see that the moment as to the origin of the stress on a 
portion of the boundary will be VpSJJv and the total 
moment which must vanish, considering S as constant, is 

ffVpZdv = fffVpttdv, 
hence 

FpHv = = €(S). 

We see therefore that S is self-conjugate. 

EXAMPLES 

(1). Purely normal stress, hydrostatic stress. In this case 
S is of the form pS = gp, where g is + for tension, — for 
pressure, and is a function of p (scalar, of course). 

(2). Simple tension or pressure. 

H = — paSa. 
(3). Shearing stress. 

H = - p(aSp + PSa), 
|S not parallel to a. 



DEFORMABLE BODIES 271 

(4). Plane stress. 

8 - giaSa + g 2 (3S(3. 
(5). Maxwell's electrostatic stress. 

H= l/87r-FvP()VP, 

where P is the potential. 

28. The quadric Spap = — C is called the stress quadric. 
Its principal axes give the direction of the principal stresses. 
Since Sp is the direction of the normal we may arrive at a 
graphical understanding of the stress by passing planes 
through the center, and to each construct the conjugate 
diameter. This will give the direction of the stress, and 
since Tap is inversely proportional to the perpendicular 
from the origin on the tangent plane at p, if we lay off on 
the conjugate diameter distances inversely as the per- 
pendiculars, we shall have the vector representation of the 
stress. When the diameter is normal to its conjugate plane, 
there will be no component of the corresponding vector 
that is parallel to the plane, that is, no tangential stress. 
Such planes will be the principal planes of the stress. 

It is evident that a stress is completely known when the 
self-conjugate linear vector function H is known, which 
depends therefore upon six parameters. We shall speak, 
then, of the stress H, since H represents it. This proposi- 
tion is sometimes stated as follows: stress is not a vector 
but a dyadic (tensor). From this point of view the six 
components of the stress are taken as the coordinates of a 
vector in six-dimensional space. These components in the 
quaternion notation are, for a, (3, y, a trirectangular system, 

- SaXa, - S/3E0, - SyZy, - Saafi = - S(3Za, 
- SpEy = - SyZp, - SyZa = - Sa3y. 



272 VECTOR CALCULUS 

That is, 

X x Y y Z t , Xy = Y X i Y t = Zy, Zi x ~ X 2. 

It is easy to see now that certain combinations of these 
component stresses are invariant. Thus we have at once 
the three invariants mi, m 2 , m 3 , which are 

X x ~r* Yy~\~ %zt YyZz -f- ZgX x ~r X x Y y — Y z — Z x — X y , 
X t Y yZ z -\- ZXyY Z Z X X X Y z Y y Z x Z z X y . 

For any three perpendicular planes these are invariant. 

EXERCISE 

What are the principal stresses and principal planes of the five ex- 
amples given above? 

29. Returning to the equation of a small displacement, 
we may write it 

er" = i + <T l EV. 

Hence the time rate of storage or dissipation of energy is 

W'=- fffSa'Zvdv. 

The other terms of the kinetic energy are not due to storage 
of energy. 

Now we have an experimental law due to Hooke which in 
its full statement is to the effect that the stress dyadic is a 
linear function of the strain dyadic. The latter was shown 
to be 

<Po= -^S()V-<7+ V&rOJ. 

The law of Hooke then amounts to saying that S is a linear 
function of a and V where V operates upon a, and owing 
to the self-conjugate character of <p, we must be able to 
interchange V and a, that is, 

S = 6[(), V, a}. 



DEFORMABLE BODIES 273 

First, it follows that if the strain <p is multiplied by a 
variable parameter x, that the stress will be multiplied by 
the same parameter. We have then for a parametric change 
of this kind which we may suppose to take place in a alone 
a' = ax' . Hence for a gradually increasing a, we would 
have 

W = - xx'fffSaSVdv, 

w = - iyyy&rEv &% 

if x runs from to 1. This gives an expression for the 
energy if it is stored in this special manner. If the work is 
a function of the strain alone and not dependent upon the 
way in which it is brought about, W is called an energy- 
function. It is thus seen to be a quadratic function of the 
strain. In case there is an energy function, we have for 
two strain functions due to the displacements cr lf a 2 

Si = e[(), en, Vi], H 2 = G[(), o- 2 , v 2 ]- 

The stored energy for the two displacements must be the 
same either way we arrange the displacements, hence we 
have 

So- 2 e 3 [V3, *i» Vi] = (Scr 1 e 4 [V*i <r 2 , V 2 ], 

where the subscripts 3, 4 merely indicate upon what V acts. 
This is equivalent to saying that so far as vector function 
is concerned, in the form 

SaG[(3, 7, 5] 

we can interchange a, (3 and y, 5. Since S is self-conjugate, 
is self-conjugate, and we can interchange a and (3. From 
the nature of the strain function we can interchange y, 8. 
Of course, in the forms above we cannot interchange the 
effect of the differentiations. 



274 VECTOR CALCULUS 

We have in this way arrived at six linear vector functions 

<P\l <f22 <P32 <f23 <fn <Pl2> 

wherein we can interchange the subscripts, and where 
<Pn = 0[Q,a,a] ••• ^23= 6[(),ft7] v\, 

a /3 7 being a trirectangular system of unit vectors. We 
have further a system of thirty-six constituents Cmu c n i2, 
• • • where 

Cim = — Sa<pn<x, C1112 = — Sa<pn<x, • • •, 

each of the six functions having six constituents. These 
are the 36 elastic constants. If there is an energy function, 
they reduce in number to only 21, for we must be able to 
interchange the first pair of numbers with the last pair. 
There are thus left 

3 forms emu 6 of em%, 3 of Cim, 3 of C1212, 3 of C2311, 3 of 02m. 

In theories of elasticity based upon a molecular theory 
and action at a distance six other relations are added to 
these reducing the number of elastic constants to 15. These 
relations are equivalent to an interchange of the second 
and third subscript in each form, thus Cim = Ci2is- These 
are usually called Cauchy's relations, but are not commonly 
used. (See Love, Elasticity, Chap. III.) 

Remembering the strain function <p , we can interpret 
these coefficients with no difficulty, for we have 

— SaipoCXj • fty, 

the stress dyadic due to the strain component — Sa&oaj, 
where a;, a ; are any two of the three a, (3, y. cijki is the 
component of the stress across a plane normal to otj in the 
direction a t due to the strain component — Sak<Poai- 



DEFORMABLE BODIES 275 

EXAMPLES 

(1). If Sij = — Soti<pocxj, show that we have for the energy 
function 

W = ^CnnSn + 2cii22SnS 2 2 + i^c 12 i 2 s 12 2 

+ 201223^12^23 + SCni2*ll*l2 + 2Cii 2S S n S 23 . 

(2). When there is a plane of symmetry, say in the direc- 
tion normal to 7, all constants that involve 7 an odd number 
of times vanish, for the solid is unchanged by reflection in 
this plane. Only thirteen remain. If there are two per- 
pendicular planes of symmetry, normal to (3, y, the only 
constants left are of the types 

ClUli C1122, Ci212j 

the plane normal to a is thus a plane of symmetry also. 
There are nine constants. This last case is that of tesseral 
crystals. 

(3). If the constants are not altered by a change of a into 
— a, (3 into — (3, as by rotation about 7 through a straight 
angle, then the plane normal to 7 is a plane of symmetry. 

(4). Discuss the effect of rotation about 7 through other 
angles. 

(5). When the energy function exists we have 

0(X, fi, v) - 90*, X, v) = - VvQV\\x, where 6' = 6. 

30. A body is said to be isotropic as to elasticity when the 
elastic constants are not dependent upon directions in the 
body. In such case the energy function is invariant under 
orthogonal transformation. It must, therefore, be a function 
of the three invariants of <po, i»i, ra 2 , m 3 . The last is of 
third degree, while the energy function is a quadratic 
and therefore can be only of the form 

W = - Pmi + Am? + Bm 2 . 



276 VECTOR CALCULUS 

P is zero except for gases and is then positive. The con- 
stant A refers to resistance to compression, and is positive. 
B is a constant belonging to solids. 

The form given the quadratic terms by Helmholtz is 

Am x 2 + Bm 2 = iHm l 2 + £C[2mi 2 - 6m 2 ]. 

The [] is the sum of the squares of the differences of the 
latent roots of <po. The constant H refers to changes of 
volume without change of form, and in such change it is the 
whole energy, for if there is no change of form, the roots 
are all equal and the other term is zero. C refers to changes 
of form without change of volume, since it vanishes if the 
roots are equal and is the whole energy if there is no cubical 
expansion m\. For perfect fluids C = 0. 
The form given by Kirchoff is 



Km^tpo 2 ) + Kdrm 2 . 
From which we have 

B-C = 2KB, 3C = 2K, H= 2K(d + |), C = \K. 
We may write for solids, liquids, and gases 

W = Rdm? + Kmifao*) - Pm x . 
Later notation gives 2K6 = X, K = /x, that is, 

W = |Xmi 2 + iirriiicpo 2 ) — Pm\. 

The constants X, \x are the two independent constants of 
isotropic bodies. 

We now have for the stress function in terms of the strain 
function 

S = Xrai -f- 2/i^o. 

EXAMPLES 
(1). In the case of a simple dilatation we know S = p 



DEFORMABLE BODIES 277 

and we have for <po 

<Po= - JOSOV-ap + ASapQ) = a(). 

Substituting in the equation above, we have 

()p m X(3o) + 2 M o(). 

The cubical dilatation is thus 

3a = p/(X + |m) = p/», 

where A: is called the modulus of cubical compression. 
(2). For a simple shear 

<p, = - a/2-[aSPQ + g&xOL ™i = 0> 
S = - a/z[«<Sj8() + 0&*()]. 

If the tangential stress is T, then T = a/j,. M is the shear 
modulus or simple rigidity. 

(3). If a prism of any form is subject to tension T uniform 
over its plane ends, and no lateral traction, we have 

S = - afSaQ - Xm + 2n<p . 

From this equation, taking the first scalar invariant of 
both sides, 

T = 3mA + 2muh 
so that 

rrn= T/(3\+2fi). 

Substituting, we have 



2/i v 2ju(3X +2ju) 

We write now E = /x(3X + 2/x)/(X + /x)> the quotient of a 
simple longitudinal tension by the stretch produced, and 
called Young's modulus. Also we set 

s = X/(2X + 2/x), Poisson's ratio, 



278 VECTOR CALCULUS 

the ratio of the lateral contraction to the longitudinal 
stretch. 

It is clear that if any two of the three moduli are known, 
the other may be found. We have 

X = E/[(l + *)(1 - 2*), M - \Ej(X + *), 
k - IE/(1 - 2s). 

In terms of E and s we have 

t»i(S)'« 



po 



-m* 



E 



(4). If | < s, k < 0, and the material would expand under 
pressure. If s < — 1, W would not be positive. 

(5). If Cauchy's relations hold, s = \ and X = /x. For 
numerical values of the moduli see texts such as Love, 
Elasticity. 

31. Bodies that are not isotropic are called aelotropic. 
For discussion of the cases and definitions of the moduli, 
see texts on elasticity. 

32. There is still the problem of finding a from cp after 
the latter has been found from S. This problem we can 
solve as follows: 

<t = <tq-\- fp^da = (To — J£<rS Vdp, where V acts on a 
= o-o + fgW&P ~ hVdpVVv] 
= *o + fgWdp ~ WiPi ~ p)VVd<r 

-d-V( Pl - p)VV<r] 
= <to- Wifii ~ Po)VVao+ f P P Modp 

-iVQ>i-p)VvM 

= <ro- \Vifii - Po)VV<ro + f P S l [<Podp 

- V( Pl - p)W<Po'dp]. 

We are thus able to express a at any point pi in terms of the 



DEFORMABLE BODIES 279 

values at p of cr, VVc, and the values along the path of 
integration of <p and FV^oO- 

EXAMPLES 

(1). Let us consider a cylinder or prism which is vertical 
with horizontal ends, the upper being cemented to a hori- 
zontal plane. Then we have the value of 

% = — gcySypSyQ, y vertical unit, 

where the origin is at the center of the lower base. The 
conditions of equilibrium are 

S V + c£ =0, or c{ * - gey, J = - gy. 

That is, the condition is realizable by a cylinder hanging 
under its own weight. The tension at the top surface is 
gel where I is the length. 

Solving for the strain, we have 

Let a = gcs/E, b = gc(l + s)IE, and note that 

FWoO = - aVy() - bVyySyQ = - aVy(). 

The integral is thus 

°"o — hV(fti — p )e 

+ fp'oiaSyp-dp + bySypSydp + aV( Pl - p)Vydp] 
= (To— \V(p\ — po)e 

+ Jl?[aSyp-8p + bySypSydp 

+« VpiVydp — adpSyp + aySpdp] 
= a — W{pi — p )e + HbyS 2 yp 

+ aV Pl Vyp + haypX, 

the differential being exact. This gives us as the value of 
a at pi, 



280 VECTOR CALCULUS 

*l « f + V(pi - p )(ieo + aVypo) + iaFprypi 

-f \byS 2 yp> f > « constants. 

Substituting a and 6, and constructing 

<Po = - J[S()V-«r+ V&r()], 

we easily verify. If the cylinder does not rotate, we may 
omit the second term and if the upper base does not move 
laterally, then the vector f reduces to — ^gcP/E-y, and 
we have 

' = - hgcP/E-y + gcs/2E-Vpyp + gc(l + s)/2E-yS 2 py. 

A plane cross-section of the cylinder is distorted into a 
paraboloid of revolution about the axis and the sections 
shrink laterally by distances proportional to their distances 
from the free end. 

(2). If a cylinder of length 21 is immersed in a fluid of 
density c', its own density being c, the upper end fixed, p 
the pressure of the fluid at the center of gravity, then we 
have the stress given by 

H = - (p + gc'Syp) - g(c - c')(l - Syp)ySy, 

whence calculating <p , we have 

<p = 1/E-l- (p + gc'Syp)(- 1 + 2*) - gs(c - c') 

X (1 - Syp)] - ySy[g(c - c')(l - Syp)l + s)]/E. 
And 

a = f + Vdp + p[(- 1 + 2*)p - ^/(c - cO. 
- Spyg[ce - s(c + c')]/E 

+ 7lh(c ~ c')(l + s)(l - Syp) 2 ' 

+ hgp 2 W-s(c+c')]/E. 

(3). What does the preceding reduce to if c = a'? Solve 
also directly. 



DEFORMABLE BODIES 281 

(4). If a circular bar has its axis parallel to y, and the 
only stress is a traction at each end, equivalent to couples 
of moment \ira*pt, about the axis of y, a being the radius, 
that is, a round bar held twisted by opposing couples, we 
have 

S = - lidfySnO + VpySyQ), 
<Po= - HiySpyO + VpySyQ], 
a = tVpySyp. 

Any section is turned in its own plane through the angle 
— tSyp. t is the angular twist per centimeter. 

(5). The next example is of considerable importance, as 
it is that of a bar bent by couples. The equations are 

g = - E/R-Sap-ySyQ, 

Po - - (1 + s)/R-Sap-ySyQ ~ s/R-Sap-Q, 
a = iR-i-al&yp + sS 2 ap - sS 2 yap] 

+ sR~ 1 yaS(3pSap — R~ 1 ySapSyp. 

If the body is a cylinder or prism of any shape with the 
axis y horizontal, there is no body force nor traction on the 
perimeter. The resultant traction across any section is 

ff- EjR-SapdA, 

which will equal zero if the origin is on the line of centroids 
of the sections in the normal state, that is, the neutral axis. 
Thus the bar is stressed only by the tractions. at its terminal 
sections, the traction across any section being equivalent 
to a couple. 

The couple becomes one with axis (3 = ya and value 
EI/R, where 7 is the moment of inertia about an axis 
through the centroid parallel to (3. The line of centroids 
is displaced according to the law 

- Saa = iR-'S'-yp, 

19 



2N2 VECTOR CALCULUS 

so that it is approximately the arc of a circle of radius R. 
The strain-energy function is \ER~ 2 -S 2 ap, and the potential 
energy per unit length %EI/R 2 . 

For further discussion see Love, p. 127 et seq. 

(6). When E = - E-Syp-OQ, where dy = 0, and 6 = 0', 
and a may not be a unit vector, show that 

<Po = ~ (1 + 8)Syp-6Q + sSyp-mi(0), 
a = (1 + 8)tiSp6p - OpSpy] + mi«[- \yp 2 + pSpy]. 

See Love, pp. 129-130. 

33. We recur now to the equation of equilibrium 

EV + cf - 0. 

In this we substitute the value of 

H = Xmi + 2/^o = - XSVo- - (o-/S() V + V&r()), 

whence 

XV*SVcr + m W + n\/SS7<r - cf = 0, 
or 

(X + M ) VSVo- + M V 2 c - c? = 0, 

or equally since 

VV = VSVa + VVVa, 
(X + 2 M ) V»SVcr + fiVVVcr - c£ = 0. 

This is the equation of equilibrium when the displacement 
and the force £ are given. In the case of small motion we 
insert on the right side instead of 0, — ca". The traction 
across a plane of normal v is 

— (X + iJ,)vSVcr — pV\Jvv, 

where v is constant. Operating on the equilibrium equa- 
tion by *SV(), we see that 

(X+2/z)V 2 SV<r-oSv£= 0. 



DEFORMABLE BODIES 283 

If then there are no body forces £ or if the forces £ are 
derivable from a force-function P and V 2 P = throughout 
the body, we see that 

SVa 

is a harmonic function. Since rai(E) = Skmi, we see that 
mi(H) is also harmonic. 
Again we have 

(X + m)V#Vo- = - M VV, 

whence we can construct the operators 

(X + /xj V£v()£V<r - - mV 2 V&j - - M vvsv(). 
and adding the two, 

2(X + M)VSvSV(r() - - mV 2 (^V() + V&r()) 
Now we have 

g = - \SVct - m(^V() + V-ScrO), 
and since S\7<r is harmonic 

V 2 H = - /xV 2 (^V() + V&r()) = 2(X + /*) ViS ViSVcrO 
2(X + M ) 



3& 
or 



V#V£V<7() = (1 + s^VSvSVtrQ. 



V 2 H = ^- ViSvifiO. 

This relation is due to Beltrami, R. A. L. R., (5) 1 (1892). 

EXAMPLE 
Maxwell's stress system cannot occur in a solid body 
which is isotropic, free from the action of body forces, and 
slightly strained from a state of no stress, since we have 
-Wil(E) = 1/8tt-(vP) 2 , 



284 VECTOR CALCULUS 

which is not harmonic. (Minchin Statics, 3d ed. (1886), 
vol. 12, ch. 18.) 

34. We consider now the problem of vibrations of a solid 
under no body forces, the body being either isotropic or 
aeolotropic. 

The equation of vibrations is 

c<r" = 6( V, V, <r), where S = 6[(), V, <r] as before, and 

a is a function of both t and p. If the vector co represents 
the direction and the magnitude of the wave-front, the 
equation of a plane-wave will be 

u = t — Sp/co, 

since this represents a variable plane moving along its 
own normal with velocity w. By definition of a wave-front 
the displacement from the mean position is at any instant 
the same at every point. That is, a is a function of u 
and t, hence 

Vo" = — VSp/ooda/du = uT^a/du, 

and any homogeneous function of V as/(V) gives 

/V-<r = f{oT l )d n (rldu n , 

where n is the degree of /. 

The equation above for wave-motion then is 

cv" = e[oj-\ or 1 , d 2 a/du 2 ]. 

If the wave is permanent, a involves t only through u and 
if the vibration is harmonic of frequency p, 

<r" = du 2 a/d 2 = - fa. 
Therefore 

e[Uu, Uw, a] = ctrT*u. 

Hence for a plane wave propagated in the direction Uoj 



DEFORMABLE BODIES 285 

the vibration is parallel to one of the invariant lines of the 
function 

e[U<a, Uco, ()]. 

The velocity is the square root of the quotient of the latent 
root corresponding, by the density. There may be three 
plane-polarized waves propagated in the same direction 
with different velocities. The wave- velocity surface is 
determined by the equation 

S[e(w-\ co" 1 , a) - ca][e(u-\ co" 1 , (3 - cjSHeC&T 1 , co" 1 , y] = 0, 

that is, by the cubic of Q[Uu, Uu, ()]. 

If there is an energy function, Q[Uu, Uu, ()] is self- 
conjugate as may easily be seen. In such case the invariant 
lines are perpendicular, that is, the three directions of 
vibration, U 6 2 , 03, for any direction of propagation are 
mutually trirectangular. Since W is essentially positive, 
the roots are positive, and there are thus three real velocities 
in any direction. 

If g is a repeated root, there is an invariant plane of 
indeterminate lines and the condition for such is 

V[e(«T\ to" 1 , a) - ca][e(^-\ co" 1 , 0) - cfi] = 0, 

a and /3 arbitrary. There is a finite number of solutions to 
this vector equation, giving co, and these give Hamilton's 
internal conical refraction. The vectors terminate at 
double points of the wave-velocity surface. 

The index-surface of MacCullagh, that is, Hamilton's 
wave-slowness surface, is given by 

5[0(p, p, a) - ca][G(p, p, (3) - cj8][0(p, p, 7) ~ ey] = 0, 

a, jS, 7 arbitrary, which is the inverse of the wave-velocity 
surface, p is the current vector of the surface, just as co 
for the other surface, the equation being formed by setting 



286 VECTOR CALCULUS 

p = — a> -1 . The wave-surface, or surface of ray- velocity, 
is the envelope of Sp/o) = 1, or Spp = — 1, where 
/x = — w _1 . The condition is that given by the equations 
of the two other surfaces. It is the reciprocal of the index 
surface with respect to the unit sphere p 2 = — 1, or the 
envelope of the plane wave-fronts in unit time after passing 
the origin, or the wave of the vibration propagated from the 
origin in unit time. The vectors p that satisfy its equation 
are in magnitude and direction the ray- velocities. When 
there is an energy function, this ray-velocity is found 
easily, as follows: 

The wave-surface is the result of eliminating between 

0(/x, p, a) = ca, 

Q(dp, p, a) + 0(ju, dp, a) + 0(ju, /x, da) = cdcr, 
Sup = - 1-Spdfi= 0. 

From the second equation 

2SdfxG(<T, a, n) + SdaOiii, fi, a) = cSadX, 

or by the equations 

Sdp.e(<r, a, /x) = 0. 

Hence as dfi is perpendicular to p, we have 

G(<r, <r, p) = xp. 

Operate by Sp and substitute the value of x, 

Q(U<t, Ua, p) = cp. 

This equation with 6(p,p, a) = ca gives all the relations 
between the three vectors. See Joly, p. 247 et seq. 



CHAPTER XI 
HYDRODYNAMICS 

1. Liquids and gases may be considered under the com- 
mon name of fluids. By definition, a perfect fluid as dis- 
tinguished from a viscous fluid has the property that its 
state of stress in motion or when stationary can be con- 
sidered to be an operator which has three equal roots and 
all lines invariant, thus 

E = -p(), 

where p is positive, that is, a pressure, or S = —p. If the 
density is c, we have, when there are external forces and 
motion, the fundamental equation of hydrodynamics 

<r" = J - c~ l Vp. 

In the case of viscous fluids we have to return to the 
general equation 

c (*" - {) « - Vp - (X + m) VSV o- -mW. 

2. When there is equilibrium 

Vp = c£. 

If the external forces may be derived from a force function, 
P, we have Vp = cVP, hence — SdpVp = — ScdpVP, 
or dp = cdP for all directions. That is, any infinitesimal 
variation of the pressure is equal to the density into the 
infinitesimal variation of the force function. In order that 
there may be equilibrium under the forces that reduce to 
£, we must have £ subject to a condition, for from Vp = c£, 
we have V 2 p = Vc£ + cV£, whence ££V£ = 0, and 
VV% = F£Vlogc. 

287 



288 VECTOR CALCULUS 

If £ = VP, the condition is, of course, satisfied, and 
from the last equation we see that £ is parallel to Vc, that 
is to say, £ is normal to the isopycnic surface at the point, 
or the levels of the force function are the isopycnic surfaces. 
The equation Vp = c£ states that £ is also a normal of the 
isobaric surfaces. In other words, in equilibrium the iso- 
baric surfaces, the isopycnic surfaces, and the isosteric sur- 
faces are geometrically the same. However, it is to be 
noted that if a set of levels be drawn for any one of the 
three so that the values of the function represented differ 
for the levels by a unit, that is, if unit sheets are constructed, 
then the levels in the one case may not agree with the levels 
in the other two cases in distribution. 

The fundamental equation above may be read in words: 
the pressure gradient is the force per unit volume. Specific 
volume times pressure gradient is the force per unit mass. 

We can also translate the differential statement into 
words thus: the mean specific volume in an isobaric unit 
sheet is the number of equipotential unit sheets that are in- 
cluded in the isobaric unit sheet. The average density in an 
equipotential unit sheet is the number of isobaric unit sheets 
enclosed. 

Since dp and dP are exact differentials, we have : 

Under statical conditions the line integral of the force of 
pressure per unit mass as well as the line integral of the force 
from the force function per unit volume are independent of the 
path of integration and thus depend only on the end points. 

3. There is for every fluid a characteristic equation which 
states a relation between the pressure, the density, and a 
third variable which in the case of a gas may be the tempera- 
ture, or in the case of a liquid like the sea, the salinity. 
Thus the law of Gay-Lussac-Mariotte for a gas is 

p = const -c (1+ t^t T) f° r constant volume. 



HYDRODYNAMICS 289 

The characteristic equation usually appears in the form 
pa = RT, where in this case a is the specific volume, the 
equation reading 

dP = adp. 
From this we have 

dP = RTdp/p. 

If T is connected with p by any law such as that given 
above, we can substitute its value and integrate at once. 
Or if T is connected with the force function P by an equa- 
tion, we can integrate at once. 

Example. 
In the case of gravity and the atmosphere, suppose 
that the temperature decreases uniformly with the equi- 
potentials. Since we must in this case take P so that 
VP will be negative, we have 

dP = - RTdp/p, T = T - bP, 
whence 

dP = -dT/b, dT/T = Rbdp/p, T = T (p/p ) bR . 

Or again 

dP/(T -bP)= - R dp/p, 1 - bP/To = (p/po) R . 

We thus have the full solution of the problem, the initial 
conditions being for mean sea-level, and in terms of a or 
c as follows : 

T= T (p/p ) bR , a= a (p/po) bR -\ 

p = b-iT [i- (p/p n 

T= T (l - bTo-'P), c= c (l - To-'bP) »" 1 »" 1 - 1 , 
p = Pod- To-'bP)*- 1 *- 1 . 

Absolute zero would then be reached at a height where the 



200 VECTOR CALCULUS 

gravity potential would be 

P = To/b, 

and substituting we find c = 0, p = 0. If b is negative, 
the fictive limit of the atmosphere is below sea-level. For 
values of bR from oo to 1, for the latter value b = 0.00348 
(that is, a temperature drop of 3.48° C. per 100 dynamic 
meters of height), we have unstable equilibrium, since 
from the equations above for c we have increasing density 
upwards. The case bR = 1 is extreme; however, it is 
mathematically interesting from the simplicity that re- 
sults. Pressure and temperature would decrease uniformly 
and we should have a homogeneous atmosphere. This 
condition is unstable and the slightest displacement would 
continue indefinitely. Values of bR less than 1 lead still 
to unstable equilibrium, the state of indifferent equilibrium 
occurring when the adiabatic cooling of an upward moving 
mass of air brings its temperature to that of the new levels. 
For dry air this occurs for bR = 0.2884 = (1.4053 
- 1) /1.4053, or a fall of 1.0048° C. per dynamic hectom- 
eter. 

See Bjerknes, Dynamic Meteorology and Hydrography. 

4. The equation when there is not equilibrium gives us 

aVp — £ * — a". 

Let £ = VP, and operate by V*V (), then 

WaVp = - VV<r". 

If we multiply by SUv and integrate over any surface nor- 
mal to Up, we have 

SfSUvWaVp = - ffSUvW" = - fSdpa". 

The right-hand side is the circulation of the acceleration 
or force per unit mass around any loop, the left-hand side 



HYDRODYNAMICS 291 

is the surface integral of WaVp over the area enclosed. 
If then we suppose that in a drawing we represent the iso- 
bars as lines, and the isosterics also as lines that cut these, 
drawing a line for the level that bounds a unit sheet in each 
case (and noticing that in equilibrium the lines do not in- 
tersect), we shall have a set of curvilinear parallelograms 
representing tubes. The circulation of the force per unit 
mass around any boundary will then be the number of 
parallelograms enclosed. It is to be noticed that the areas 
must be counted positively and negatively, that is, the 
number of tubes must be taken positive or negative, ac- 
cording to whether Vfl, Vp, the two gradients, make a 
positive or a negative angle with each other in the order as 
written. This circulation of the force per unit mass may be 
taken as a measure of the departure from equilibrium. 
In the same way we find that if we draw the equipotentials 
and the isopycnics, we shall have the number (algebraically 
considered) of unit tubes in any area equal to the circula- 
tion of the force per unit volume around the bounding 
curve. 

If we choose as boundary, for example, a vertical line, an 
isobaric curve, a downward vertical, and an isobaric curve, 
the number of isobaric-isosteric tubes enclosed gives the differ- 
ence between the excess up one vertical of the cubic meters 
per ton at the upper isobar over that at the lower isobar and 
the corresponding excess for the other vertical. If the lines 
are two verticals and two equipotentials, the number of 
isopotential-isopycnic tubes is the difference of the two 
excesses of pressure at the lower levels over pressure at the 
upper levels. These are the circulations around the bound- 
aries of the forces per unit mass or unit volume as the case 
may be. 

5. If we integrate the pressure over a closed space inside 



292 VECTOR CALCULUS 

the fluid, we have 

ffyUvdA = fffVpdv = fffc&v. 

But this latter integral is the total force on the volume 
enclosed. This is Archimedes' principle, usually related 
to a body immersed in water, in which case the statement 
is that the resultant of all the pressure of the water upon 
the immersed body is equal to the weight of the water dis- 
placed. If we were to consider the resultant moment of 
the normal pressures and the external forces, we would 
arrive at an analogous statement. The field of force, how- 
ever, need not be that due to gravity. 

EXERCISE. 

Consider the case of a field in which there is the vertical 
force due to gravity and a horizontal force due to centrif- 
ugal force of rotation. 

6. We turn our attention now to moving fluids. A 
small space containing fluid with one of its points at po 
may be followed as it moves with the fluid, always con- 
taining the same particles. It will usually be deformed in 
shape. The position p of the particle initially at p will 
be a function of p and of t, say 

p = (p , t). 

The particle initially at p + dp will at the same time t 
arrive at the position 

p + dip = 6 (p + dp, t) = p — SdipVo-p, 

hence dip becomes at time t 

— SdipVo'P = <pdipo. 

It follows that the area Vdipd 2 p = V(pdip (pd 2 po, and the 



HYDRODYNAMICS 293 

volume 

— Sdipdtpdzp = — S(pdipo(pd2po<pd s po = 

— Sdip Q d2Pod d p ' m s ((p) . 

If the fluid has a constant mass, then we must have 

cdv = Codvo, or cra 3 = c . 

This is the equation of continuity in the Lagrangian form. 
The reference of the motion to the time and the initial con- 
figuration is usually called reference to the Lagrangian 
variables. 
7. Since 

dp = — SdpVp = — S<pdp Vp 

= — Sdpo<p'Vp = — SdpoVoP, 
VoP = <p'Vp = - VoSpV-p. 

But the equations of motion are already given in the form 

aVp = £ - p", 

hence in terms of the variables po and t we have 

aVop = <p'(p — p")- 

This equation, the characteristic equation of the fluid 

F(p, c, T) - 0, 

and the equation of continuity, give us five scalar equations 
expressing six numbers in terms of p and t. In order to 
make any problem definite then, we must introduce a 
further hypothesis. The two that are the most common 
are 

(1) The temperature is constant, if T is temperature, 
or the salinity is constant, if T is salinity. In case both 
variables come in, we must have two corresponding hypoth- 
eses: 



21)4 VECTOR CALCULUS 

(2) The fluid is a gas subject to adiabatic change. 
The relation of pressure to density in this case is usually 
written 

p = kc y . 

y is the ratio of specific heat under constant pressure to 
that under constant volume, as for example, for compressed 
air, 7 - 1-408. 

8. In the integrations we are obliged to pay attention 
to two kinds of conditions, those due to the initial values of 
the space occupied by the fluid at t = 0, the pressure p 
and density c , or specific volume a , at each point of the 
fluid, and the initial velocities of the particles p ' at p . 
The other conditions are the boundary conditions during 
the movement. As for example, consider a fluid enclosed 
in a tank or in a pipe or conduit. The velocity in the 
latter case must be tangent to the walls. If we have the 
general case of a moving boundary for the fluid, then its 
equation would be 

/(P, t) = o. 

If then p' is the velocity, we must have 

- Sd P Vf+ (df/dt)dt = 0, or - £p'V/+ df/dt = 0. 

If there is a free surface, then the pressure here must be 
constant, as the pressure of the air. In order to have 
various combinations of these conditions coexistent, it is 
necessary sometimes to introduce discontinuities. 

9. If we were in a balloon in perfect equilibrium moving 
along with one and the same mass of air, the barograph 
would register the varying pressures on this mass, the ther- 
mograph the varying temperatures, and if there were a 
velocitymeter, it would register the varying velocity of the 
mass. From these records one could determine graphically 
or numerically the rates of change of all these quantities as 



HYDRODYNAMICS 295 

they inhere in the same mass. That is, we would have the 
values of 

dp/dt, dT/dt, dp/dt. 

These may be called the individual time-derivatives of the 
quantities. As the balloon passed any fixed station the 
readings of all the instruments would be the same as instru- 
ments at the fixed stations. But the rates of change would 
differ. The rates of change of these quantities at the same 
station would be for a fixed p and a variable t, and could 
be called the local time-derivatives, or partial derivatives. 
They can be calculated from the registered readings. The 
relation between the two is given by the equation 

d/dt = d/dt - Sp'V. 

Thus we have between the individual and the local values 
the relations 

The last equation gives us the individual acceleration 
in terms of the local acceleration and the velocity. From 
the fundamental equation we have 

ovp = f - dp' let + w \p' = i - d p'l dt ~ *<j>% 

where the function 

0=-S()V-p', 0'=-VV(), 0o = 

K-sovy-wo), 

2 e = FVp'. 
This statement of the motion in terms of the coordinates of 



296 VECTOR CALCULUS 

any point and the time is the statement in terms of Eulers 
variables. 

Since near po, p = po + po'dt, we have the former 
function <p at this point in the form 

<p= - S()Vo-p = l + <ft(- S()V-p') = 1 + d^atpo. 

Whence 

m 3 (<p) = 1 + dtmi(6) = 1 + dt{- SVp'). 

Since the initial point is any point, this equation holds for 
any point and we have the equation of continuity in the 
form 

c - cdtSVp' = c = c + dt-dc/dt(l ~ dtSVp'), 

or, dropping terms of second order, 

dc/dt - cSVp' = 0. 

This is the* equation of continuity in the Euler form. 
If we use local values, 

dc/dt- SV(cp') = 0. 

That is, the local rate of change of the density is the con- 
vergence of specific momentum. It is obvious that if the 
fluid is incompressible, that is, if the density is constant, 
then the velocity is solenoidal. If the specific volume at a 
local station is constant, then the specific momentum is 
solenoidal. If the medium is incompressible and homo- 
geneous, then both velocity and specific momentum are 
solenoidal vectors. It is clear also that in any case the 
normal component of velocity must be continuous through 
any surface, but specific momentum need not be. If any 
boundary is stationary, then both velocity and specific 
momentum are tangential to it. 



HYDRODYNAMICS 297 

In the atmosphere, which is compressible, specific mo- 
mentum is solenoidal, but in the incompressible hydro- 
sphere, both velocity and specific momentum are solenoidal. 
Of course the specific volume of the air changes at a 
station, but only slowly, so that the approximate statement 
made is close enough for meteorological purposes. 

If at any given instant we draw at every point a vector 
in the direction of the velocity, these vectors will determine 
the vector lines of the velocity which are called lines of 
flow. These lines are not made up of the same particles 
and if we were to mark a given set of particles at any time, 
say by coloring them blue, then the configuration of the 
blue particles would change from instant to instant as they 
moved along. The trajectory of a blue particle is a stream 
line. If the particles that pass a given point are all colored 
red, then we would have a red line as a line of flow, only when 
the condition of the motion is that called stationary. In 
this case the line through the red particles would be the 
streamline through the point. If the motion is not sta- 
tionary, then after a time the red particles would form a 
red filament that would be tangled up with several stream 
lines. 

10. In the case of meteorological observations the di- 
rection of the wind is taken at several stations simultane- 
ously and by the anemometer its intensity is given. These 
data give us the means of drawing on a chart suitably pre- 
pared the lines of flow at the given time of day and the 
curves showing the points of equal-intensity of the wind 
velocity. Of course, the velocity is usually only the hori- 
zontal velocity and the vertical velocity must be inferred. 

One of the items needed in meteorological and other 
studies is the amount of material transported. If the spe- 
cific momentum in a horizontal direction is cp r , and lines 

20 



298 VECTOR CALCULUS 

of flow be drawn, then for a vertical height dz and a width 
between lines of flow equal to dn, we will have the trans- 
port equal to Tp'dndz. Since, however, we have for prac- 
tical purposes dz = — dp, we can write this in the form 

transport = Tp'dn(— dp). 

In order to do this graphically we first draw the lines of 
flow and the intensity curves. An arbitrary outer bound- 
ary curve is then divided into intervals of arc such that 
the projection of an interval perpendicular to the nearest 
lines of flow multiplied by the value of Tp' is a constant. 
Through these points a new set of lines of flow is constructed. 
The transport between these lines is then known horizon- 
tally for a constant pressure drop, by drawing the intensity 
curves that represent Tp'dn, and if these are at unit values 
of the transport, they will divide th£ lines of flow into quad- 
rilaterals such that the amount of air transported horizon- 
tally decreases or increases by units, and thus the vertical 
transport must respectively increase or decrease by units, 
through a sheet whose upper and lower surfaces have pies- 
sure difference equal to dp = — 1. Towards a center of 
convergence the lines of flow approach indefinitely close. 
dn decreases and it is clear that the vertical transport up- 
ward increases. There may be small areas of descending 
motion, however, even near such centers. In this manner 
we may arrive at a conception of the actual movement of 
the air. 

Since the specific momentum is solenoidal, we can as- 
certain its rate of change vertically from horizontal data. 
For 

= SVcp' = — dZ/dz + horizontal convergence, 
or 

dZ/dz = horizontal convergence of specific momentum. 



HYDRODYNAMICS 299 

Substituting the value of dz, we have 

dZ/ (— dp) = horizontal convergence of velocity, 
dZ/dp = dT P 'lds+ Tp'b. 

where ds runs along the lines of flow, and 5 is the diver- 
gence per unit ds of two lines of width apart equal to 1. 

These considerations enable us to arrive at the complete 
kinematic diagnosis of the condition of the air. On this is 
based the prognostications. 

11. When the density c is a function of the pressure p, 
and the forces and the velocities can be expressed as gradi- 
ents, then'we have a very simple general case. Thus let 

c = f(p), i = V«(p, 0i p' = Vv(p, t), 

and set 

Q = u — fa&p, then VQ = £ — aVp, 

the equations of motion are 

dp'/dt + 0(p') = VQ, or since p' = Vv, 
V[dv/dt + iT 2 Vv- Q] = 0. 

Hence the expression in brackets is independent of p and 
depends only on t and we have 

dv/dt+iFW- Q = h(t). 

We could, however, have used for v any function differing 
from v only by a function of t, thus we may absorb the func- 
tion of the right into v and set the right side equal to zero. 
We thus have the equations of motion 

dv/dt + JPVfl - Q = 0, dc/dt - SV(cVv) = 0, 
c = /(p). 

From these we have v, c, p in terms of p and /. 

12. In the case of a permanent motion, the tubes of flow 
are permanent. If we can set £ = Vw(p), then we place 



300 VECTOR CALCULUS 

Q = u — fadp, and noticing that p' and Q do not depend 
on t, we have 

Sp'V-p' = - VQ. 

If we operate by — Sdp = — S(dsUp'), we have 
(kSUp'Tp'VTp' on the left, since Sp'V-Up' = 0. Hence 
from this equation we have at once 

- SdpGTV - Q) = 0. 

Hence along a tube of flow of infinitesimal cross-section 

tiy-4-a 

This is called Bernoulli's theorem. C is a function of the 
two parameters that determine the infinitesimal line of 
flow. Hence along the same tube of flow 

J(IV - TW) = Q ~ Qo = u - u - f p * adp. 

In the case of a liquid a is constant and we can integrate 
at once, giving 

}ZV- u+ap= C. 

From this we can find the velocity when the pressure is 
given or the pressure when the velocity is given. Since 
the pressure must be positive, it is evident that the velocity 
square ^ 2{u + C), or else the liquid will separate. This 
fact is made use of in certain air pumps. In the case of no 
force but gravity we have u = gz, 

iTV-g*+ap= C. 

This is the fundamental equation of hydraulics. We can- 
not enter upon the further consideration of it here. 

Vortices. 
13. In the case of p' = Vv it is evident that VVp f = 0. 
When this vector, or the vector e (§9) does not vanish, 



HYDRODYNAMICS 301 

there is not a velocity potential and vortices are said to 
exist in the fluid. It is obvious that if a particle of the 
fluid be considered to change its shape as it moves, then e 
is the instantaneous velocity of rotation. At any instant 
all the vortices will form a vector field whose lines have the 
differential equation 

VdpWp' = = SdpV' p - VSp'dp; 
that is, 

Q'dp = dp', or 0'p' = dp'jdt, 



from which 



p' . */><%'. 



These vector lines are called the vortex lines of the fluid. 
Occasionally the vortex lines may be closed, but as a rule 
the solutions of such a differential equation as the above 
do not form closed lines, in which case they may terminate 
on the walls of the containing vessel, or they may wind 
about indefinitely. The integral of this equation will 
usually contain t, and the vortices then vary with the time, 
but in a stationary motion they will depend only upon the 
point under consideration. 

14. The equations of motion may be expressed in terms 
of the vortex as follows, since 



we have 
and thus 



Vp'VVp' ' = Sp'V-p'-iVp' 2 ,. 



Sp'V-p' =2Vp'e + iVp'\ 



aVp = i - dp' Idt + JVp /2 + 2Vp'e. 



15. When now £ = \/u{p, t), and c = f(p), we set 
P = fadp, giving VP = aVp, and thence 

VP = Vu - dp' Idt + JVp /2 - 2Vep'. 



302 VECTOR CALCULUS 

Or, if we set II = u-\- Jp' 2 — P, we have 

dp'/dt + 2Ve P ' = VII. 

Operate on this with V-V(), and since VV dp'/dt 
= 2de/dt, and WVep' = SeV -p' - eSVp' - Sp'V-e, de/dt 
— Sp'V-e = de/dt, SVp' by the continuity equation is 
equal to c~ l dc/dt = — a~ l da/dt, we have 

d(ae)/dt = - S(ae)V-p' = 6(ae). 

This equation is due to Helmholtz. 

If we remember the Lagrangian variables, it is clear that 
6 is a function of the initial vector p and of t, hence the 
integral of this equation will take the form 

ae = e fm 'a,e Q = e' ~ s ^^' dt a e = ^(t)a e . 

But the operator is proved below to be equal to <p itself, 
so that when £ = Vu, 

ae = a Se Vo-p = + ao<p€ , 

or finally we have, if we follow the stream line of a particle, 
which was implied in the integration above, Cauchy's form 
of the integral 

(a/a )e = — *Se Vo'jP, 

where p is a function of p and t. It is evident now if 
for any particle e is ever zero, that is, e = 0, that always 
e = 0. This is equivalent to Lagrange's theorem that if 
for any group of particles of the fluid we have a velocity 
potential, then that group will always possess a velocity 
potential. (It is to be noted that velocity potential and 
vortex are phenomena that belong to the particles and the 
stream lines, and not to the points of space and the lines 
of flow.) It must be remembered too, that this result 
was on the supposition that the density was a function of 



HYDRODYNAMICS 303 

the pressure alone, and that the external forces £ were 
conservative. 

16. We may deduce the equation above as follows, which 
reproduces in vector form the essential features of Cauchy's 
demonstration. (Appell, Traite de Mec. Ill, p. 332.) 

Let dp/dt = a, and Q = u — fadp, then, remembering 
that Q is a function of p and t, and p is a function of p 
and /, 

da/dt = VQ(p, t). 

Also VoQ(po, t) = - VoSpVQ = — VoSpda/dt, where Vo 
operates on p only; or we can write 

VoQ = <p' da/dt. 

Hence, operating with FVo( ), we have V\7o(p f da/dt = = 
d/dt(VVo<p'a). Thus the parenthesis equals its initial 
value, that is, since the initial value of cp'a is a , and since 
Vo = <p'V, 

VVo<p'<r = 2e = V<p'Vv'<r = m z {<p)<p~ l VS7a = 2m 3 (p~ 1 e. 

Thus we have at once m 3 e = (pe . This is the same as the 
other form, since ra 3 = a/a Q . This equation shows the 
kinematical character of e, and that no forces can set up e 
or destroy it. 

17. The circulation at a given instant of the velocity 
along any loop is 

I = - fSdpp'. 

The time derivative of this is dl/dt = tf^SdpS/Sp'p' 
- Sdpp") = £(- SdpW tip' 2 - Q] ). But this is an inte- 
gral of an exact differential and vanishes. Hence if the 
forces are conservative and the density depends on the 
pressure, the circulation around any path does not change 
as the particles of the path describe their stream lines. The 



304 VECTOR CALCULUS 

circulation is an integral invariant. This theorem is due to 
Lagrange. If we express the circulation in the form 

I - ' - ffSdvVp' = - 2ffSdpe, 

we see that the circulation is twice the flux of the vortex 
through the loop. Hence as the circulation is constant, 
the flux of the vortex through the surface does not vary 
in time, if the surface is bounded by the stream loop. The 
flux of the vortex through any loop at a given instant is 
the vortex strength of the surface enclosed by the loop. 
If a closed surface is drawn in the fluid, the flux through it 
is zero, since the vortex is a solenoidal vector. 

18. If we take as our closed surface a space bounded by a 
vortex tube and two sections of the tube, since the surface 
integral over the walls of the tube is zero, it follows that 
the flux of the vortex through one section inwards equals 
that over the other section outwards. Combining these 
theorems, it is evident that the vortex strength, or wr- 
ticity, of a vortex tube is constant. Thus the collection of 
particles that make up the vortex tube is invariant in time. 
In a perfect fluid a vortex tube is indestructible, and one 
could not be generated. 

19. It is evident from what precedes that a vortex tube 
cannot terminate in the fluid but must end either at a wall 
or a surface of discontinuity, or be a closed tube with or 
without knots, or it may wind around infinitely in the fluid. 

If a vortex tube is taken with infinitesimal cross-section, 
it is called a vortex filament. 

20. We consider next the problem of determining the 
velocity when the vortex is known. That is, given e, to 
find a = p'. We consider first the case of an incompressible 
fluid, in which the velocity is solenoidal, that is, SVcr = 0. 
This with the equations at the boundaries gives us the 



HYDRODYNAMICS 305 

following problem : to find a when 2e = FVo", SVer = 0, 
SUva = at the boundaries, or if infinite a a = 0. This 
problem has a unique solution, if the containing vessel is 
simply connected. We cannot enter extensively into it, 
for it involves the theory of potential functions, and may 
be reduced to integral equations. However, since SVv = 0, 
we may set a = VVr, where *$Vr = 0, whence 

V 2 r = 2e, 

and we may suppose r is known, in the form 

T = h7ffSfe/T(p- Po )dv. 

If we operate upon this by FV( ), we find a formula for a, 

a = H,2ir-fffVe(p - p )/T\p - Po )dv. 

As we see, this formula is capable of being stated thus: 
the velocity is connected with its vortex in the same way 
as a magnetic field is connected with the electric current 
density that produces it, the vortex filament taking the 
place of the cm rent, the strength of current being Tej2ir, 
and the elements of length of the tube acting like the ele- 
ments of current. This solution holds throughout the 
entire fluid, even at points outside the space that is actually 
in motion with a vortex. 

Since the equation of the surface of the tube can be 
written in the form 

F( P , t) = 0, 

this surface will move in time. Its velocity of displace- 
ment is defined like that of any discontinuity, as 
UvFdF/dt. On one side the velocity is irrotational, on 
the other it is vortical. On the irrotational side we have 
the velocity of the form a — V?, and we must have on 



306 VECTOR CALCULUS 

that side the same velocity of displacement in the form 

UpSUpVP. 

The energy involved in a vortex on account of the velocity 
in the particles is 

K * - \cfffp' 2 dv 
= " hcfffSp'Vrdv 
= ¥fff [SV(p'r) - 2Sre]dv 
= hcffSdvp'r - cfffSredv 
= — cj J 'fSredv over all space 
= c/2T.SffSSSSee'lT(p - p )dvdv'. 

This is the same formula as that of the energy of two cur- 
rents. In the expression every filament must be considered 
with regard to every other filament and itself. 

Examples. (1). Let there be first a straight voitex fila- 
ment terminating at the top and bottom of the fluid. Let 
all the motion be parallel to the horizontal bottom. Then 

Sya = 0, Vye = 0, de/dt = 0. 

We have then 

a = VyVw, 2e = — yV 2 w = 2zy, 
say, 

w = — 7r l ffz log rdA. 

For a single vortex filament of cross-section dA and strength 
k = zdA, we have 

iv = — k/w log r = — kjir log V (# 2 + 2/ 2 ) 
a= Vy(p- po)IT>(p- p ).k/T, 

where p is measured parallel to the bottom. 

The velocity is tangent to the circles of motion and in- 
versely as the distance from the vortex filament. The 
motion is irrotational save at the filament itself. 



HYDRODYNAMICS 307 

For the effect of vortices upon each other, and their 
relative motions, see Webster, Dynamics, p. 518 et seq. 

(2). For the case of a vortex ring or a number of vortex 
rings with the same axis, see Appell, Traite, vol. Ill, p. 431 
et seq. 

21. In the more general case in which the fluid is com- 
pressible we must resort to the theorem that any vector 
can be decomposed into a solenoidal part and a lamellar 
part and these may then be found. The extra term in the 
electromagnetic analogy would then be due to a perma- 
nent distribution of magnetism as well as that arising fiom 
the current. 

EXERCISES 

1. If Sea = 0, then it is necessary and sufficient that a = M\/P, 
M being a function of p. 

2. Discuss the case Vae = 0. Beltrami, Rend. R. 1st. Lomb. (2) 
22, fasc. 2. 

3. Discuss Clebsch's transformation in which we decompose <r thus, 
o- = Vm + lVV. Show that the vortex lines are the intersections of 
the surfaces I and v, and that the lines of flow form with the vortex lines 
an orthogonal system only when the surfaces I, u, v are triply orthog- 
onal. 

4. Discuss the problem of sources and sinks. 

5. Consider the problem of multiply-connected surfaces, containing 
fluids. 

22. It will be remembered that Helmholtz's theorem 
was for the case in which the impressed forces had a poten- 
tial and the density was a function of the pressure. In 
this case we will have the equation 

da/dt + 2Vea = { - aVp + JVtf 2 . 

Operate by |FV( ) and notice that 

de/dt - eSVa - SaV-e = a - l d(ae)/dt, 

whence we have the generalized form 

a- l d(ae)ldt + SeV -<r = iVV£ - fFVaVp. 



308 VECTOR CALCULUS 

If now at the instant t the particle does not rotate and if a 
is a function of p alone, then at this instant de/dt = JFV£, 
and the paiticle will acquire an instantaneous increase of 
its zero vortex equal to the vortex of the impressed force. 
That is, £ must be peimanently equal to zero if there is to 
be no rotation at any time. 

If FV£ = but a is not a function of p alone, then we 
have 

a- 1 d(ae)/dt + SeV -<r = - §WaVp. 

The right side is a vector in the direction of the intersection 
of the isobaric and the isosteric surfaces. Now if we take 
an infinitesimal length along the vortex tube, I, the cross- 
section being A, the vorticity is ATe = m, the mass is 
cAl = constant = M. Then we have, since ae = AlejM 
= mlUejM, 

- SeV-<r = md(lUe)dtaM « - ~ fUeV* - ^^- e l f 

I I at 

a- 1 d(ae)/dt+ SeV -a = 

dmldt-lUe/aM'+ md{We)la Mdt - md(lUe)/dtaM = 

dm/dt-lUe/aM = Ve-dTe/dt = 

±| number of tubes. 

Hence the moment m of the vortex will usually change 
with the time unless the surfaces coincide. Thus a rotat- 
ing particle may gain or lose in vorticity. If then the 
isobaric and isosteric surfaces under the influence of heat 
conditions intersect, vortices will be created along the lines 
of intersections of the surfaces and these will persist until 
the surfaces intersect again, save so far as viscosity 
interfeies. 

23. Finally we consider the conditions that must be 
put upon surfaces of discontinuity, in this case of the first 
order in <r, that is, a wave of acceleration. 



HYDRODYNAMICS 309 

Let c be a function of p only. Then 

a\/p = dp/dc \7log c, and the equation of motion becomes 
p" = J — dp/dc • V log c. 

Let the equation of the surface of discontinuity be f(p , t) 
= 0, the normal v. Let £, a, p, and c be continuous as 
well as dp/dc, but p" = a' be discontinuous at the suiface. 
Then on the two sides of the surface we have the jump, 

by p. 263, 

\p"\ = - dp/dc[V log c], 
or 

G 2 ix= dp/dc -UVfSfiUVf. 

It follows, therefore, that we must have V/iUVf = and 
G = V (dp/dc), or else we have G = and SnUVf = 0. 
In the first case the discontinuity is longitudinal, in the 
second transversal. This is Hugoniot's theorem. In full 
it is: 

In a compressible but non- viscous fluid there are possible 
only two waves of discontinuity of the second order; a 
longitudinal wave propagated with a velocity equal to 
V (dp/dc), and a transversal wave which is not propagated 
at all. 

The formula for the velocity in the first case is due to 
Laplace. Also we have for the longitudinal waves [&Vo"] 
= — GSfxUVf, for transversal waves equal to zero. On 
the other hand, for longitudinal waves, [FVo\| = 0, for 
transversal, = GVUVf^. 



310 VECTOR CALCULUS 



REFERENCES. 

1. Mathematische Schriften (Ed. Gerhart). Berlin, 1850. Bd. II, 
Abt. 1, p. 20. 

2. On a new species of imaginary quantities connected with a theory 
of quaternions. Proc. Royal Irish Academy, 2 (1843), pp. 424-434. 

3. Die lineale Ausdehnungslehre. Leipzig, 1844. 

4. Gow: History of Greek Mathematics, p. 78. 

5. Ars Magna, Nuremberg, 1545, Chap. 37; Opera 4, Lyon, 1663, p. 
286. 

6. Algebra. Bologna, 1572, pp. 293-4. 

7. Om Directiones analytiske Betejning. Read 1797. Nye Samm- 
lung af det kongelige Danske Videnskabernes Selskabs Skrifter, (2) 
5 (1799), pp. 469-518. Trans. 1897. Essai sur la representation de 
la direction, Copenhagen. 

8. Essai sur une maniere de repr6senter les quantites imaginaires 
dans les constructions g6ometriques. Paris, 1806. 

9. Theoria residuorum biquadraticum, commentation secunde. 1831. 

10. Annales Math, pures et appliqu6es. 4 (1814-4), p. 231. 

11. Theory of algebraic couples, etc. Trans. Royal Irish Acad., 17 
(1837), p. 293. 

12. Ueber Functionen von Vectorgrossen welche selbst wieder Vector- 
grossen sind. Math. Annalen, 43 (1893), pp. 197-215. 

13. Grundlagen der Vektor-und Affinor- Analysis. Leipzig, 1914. 
\\. Lectures on Quaternions. Preface. Dublin, 1853. 

15. Note on William R. Hamilton's place in the history of abstract 
group theory. Bibliotheca Mathematica, (3) 11 (1911), pp. 314-5. 

16. Leipzig, 1827. 

17. Leipzig. 

18. Elements of Vector Analysis (1881-4), New Haven. Vol. 2, 
Scientific Papers. 



INDEX. 



Acceleration 27 

Action : 14, 28 

Activity 15, 129, 142 

Activity-density 15, 131 

Algebraic couple 4, 65 

Algebraic multiplication 9 

Alternating current 71 

Ampere 30 

Anticyclone 47 

Area 142 

Areal axis 198 

Argand 4 

Ausdehnungslehre 3, 9 

Average velocity 57 

Axial vector 30 

Barycentric calculus 8 

Bigelow 50, 60 

Biquaternions 3, 126 

Biradials 94 

Bivector 29 

Bjerknes 48, 57, 59, 290 

Cailler 2 

Cardan 3 

Center (singularity) 44 

Center of isogons 48 

Change of basis 54 

Characteristic equation 125 

Characteristic equation of 

dyadic 221 

Chi of dyadic 235 

Christoffel's conditions 266 

Circuital derivative 167 

Circular multiplication 9 

Circulation 78, 129 

Clifford 3, 90 

Combebiac 3 

Complex numbers 63 

Congruences 51, 138 

Conjugate 66 

Conjugate function 5 

Continuous group 195 

Continuous plane media 87 

Convergence 177 

Coulomb 13 

Couple 139 



Crystals 109 

Cubic dilatation 258 

Curl 76,82, 184 

Curl of field 77 

Curvature 148, 152 

Curves 148 

Cycle 30, 37 

Cyclone 47 

Derivative dyad 242 

Developables 150 

Dickson 105 

Differential of p . . . ' 145 

Differential of q 155, 159 

Differential of vector 55 

Differentiator 248 

Directional derivative 166 

Discharge 130 

Discontinuities 261 

Dissipation (plane) 84 

Dissipation, dispersion 180 

Divergence 76, 82 

Divergence of field 77 

Dyadic 2, 11, 218 

Dyadic field 246 

Dyname . 2 

Dyne 29 

Electric current 30 

Electric density current 30 

Electric induction 32 

Electric intensity 31, 139 

Energy 14 

Energy current 30 

Energy-density 15, 131 

Energy-density current 30 

Energy flux 142 

Equation of continuity 87 

Equipollences 71 

Equipotential 15 

Erg 14 

Euler 107 

Exact differential 190 

Exterior multiplication 9 

Extremals 160 

Eye of cyclone 47 

311 



:;il> 



VECTOR CALCULUS 



Farad 32, 73 

Faux 37, 38 

Faux-focus 44 

Feuille 30 

Feuillets 2 

Field 13 

Flow 142 

Flux 29, 130, 142 

Flux density 29 

Focus 41 

Force 29 

Force density 28, 141 

Force function 18 

Franklin 90 

Free vector 8, 25 

Frenet-vSerret formulae 148 

Functions of dyadic 238 

Function of flow 88 

Functions of quaternions. ... 121 

Gas defined 87 

Gauss 4/ 

Gauss (magnetic unit). . . .32, 130 

Gaussian operator 108 

General equation of dyadic . . 220 

Geometric curl 76 

Geometric divergence 76 

Geometric loci 133 

Geometric vector 1 

Geometry of lines 2 

Gibbs 2, 11, 215 

Gilbert 32, 130, 143 

Glissant 26 

Gradient 16, 163 

Gram 15 

Grassmann 2, 3, 9 

Green's Theorem 205 

Groups 8 

Guiot 138 

Hamilton 2, 3, 4,65,95 

Harmonics 84, 169 

Heaviside 31 

Henry (electric unit) 32, 73 

Hertzian vectors 33 

Hitchcock 49 

Hodograph 27 

Hypernumber 3, 94 

Imaginary 65 

Impedance 73 

Inductance. 73 

Inductivity 32 

Integral of vector 56 



Integrating factor 191 

Integration by parts 198 

Interior multiplication 10 

Invariant line 219 

Irrotational 88 

Isobaric 15,288 

Isogons 34 

Isohydric 15 

Isopycnic 15, 288 

Isosteric 15, 288 

Isothermal 15 

Joly 138, 147 

Joule 14 

Joule-second 14 

Kinematic compatibility .... 266 

Kirchoff's laws .' 73 

Koenig 198, 205 

Laisant 71 

Lamellae 15 

Lamellar field 84, 181 

Laplace's equation 214 

Latent equation 220 

Laws of quaternions 103 

Leibniz 3 

Level 15 

Line (electric unit) 32, 130 

Lineal multiplication 9 

Linear associative algebra ... 3 

Linear vector function 218 

Line of centers 46 

Line of convergence 47 

Line of divergence 47 

Line of fauces 46 

Line of foci 46 

Line of nodes 45 

Lines as levels 80 

Liquid defined 87 

MacMahon 75 

Magnetic current 31 

Magnetic density current 31 

Magnetic induction 32 

Magnetic intensity 32, 139 

Mass 15 

Matrix unity 65 

Maxwell 13 

McAulay 3 

Mobius 8 

Modulus 66 

Moment 138 

Moment of momentum 139 



INDEX 



313 



Momentum 28 

Momentum density 28 

Momentum of field 141 

Monodromic 14 

Monogenic 89 

Moving electric field 140 

Moving magnetic field 140 

Multenions 3 

Multiple 6 

Mutation 108 

Nabla as complex number. . . 82 

Nabla in plane 80 

Nabla in space 162 

Neutral point 47 

Node 37,38 

Node of isogons 48 

Non-degenerate equations . . . 225 

Norm 66 

Notations 

One vector 12 

Scalar 127 

Two vectors 136 

Derivative of vectors 165 

Divergence, vortex, deriva- 
tive dyads 179 

Dyadics 248 

Ohm (electric unit) 73 

Orthogonal dyadic 241 

Orthogonal transformation . . 55 

Peirce, Benjamin 3 

Peirce, B. O 85 

Permittance 73 

Permittivity 32 

Phase angle 71 

Plane fields.. 84 

Poincare 36, 46 

Polar vector 30 

Polydromic 14 

Potential ,. .. 15, 17 

Progressive multiplication ... 10 

Power 76 

Poynting vector 141 

Pressure 142 

Product of quaternions 98 

Product of several quater- 
nions 113 

Product of vectors 101 

Quantum 14 

Quaternions 2, 3, 6, 7, 95 



Radial 26 

Radius vector 26 

Ratio of vectors 62 

Reactance 73 

Real 65 

Reflections \ 108 

Refraction . . 112 

Regressive multiplication. ... 10 

Relative derivative 18 

Right versor 96 

Rotations 108 

Rotatory deviation 175 

Saint Venant's equations. . . . 260 

Sandstrom 35, 49 

Saussure 2 

Scalar 13 

Scalar invariants 220, 239 

Scalar of q 96 

Schouten 7 

Science of extension 2 

Self transverse 234 

Servois 4 

Shear 256 

Similitude 242 

Singularities of vector lines . . 244 

Singular lines 45 

Solenoidal field 84, 181 

Solid angles 117 

Solution of equations 123 

Solution of differential equa- 
tions 195 

Solution of linear equation. . . 229 

Specific momentum 28 

Spherical astronomy 110 

Squirt 90 

Steinmetz 68, 71 

Stoke's theorem 200 

Strain 253 

Strength of source or sink ... 90 

Stress 143,269 

Study . . . 2 

Sum of quaternions 96 

Surfaces 151 

Symmetric multiplication ... 9 

Tensor 65 

Tensor of q 96 

Torque 140 

Tortuosity 149 

Trajectories 150 

Transport 130,298 

Transverse dyadic 231 

Triplex 25 



314 



VECTOR CALCULUS 



Triquaternions 3 

Trirectangular biradials 100 

Unit tube 18 

Vacuity 220 

Vanishing invariants 240 

Variable trihedral 172 

Vector 1 

Vector calculus 1, 25 

Vector field 23, 26 

Vector lines 33 

Vector of q 96 

Vector potential 33, 93, 181 

Vector surfaces 34 

Vector tubes 34 



Velocity 27 

Velocity potential 18 

Versor 65 

Versor of q 96 

Virial 129 

Volt 31, 130, 143 

Vortex 92, 187, 187 

Vorticity 247,304 

Waterspouts 50 

Watt 15 

Weber 14 

Wessel 4 

Whirl 90 

Zero roots of linear equations. 230 



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